IN THE DAYS OF HIS FLESH HE OFFERED UP BOTH PRAYERS AND SUPPLICATIONS WITH LOUD CRYING AND TEARS: os en tais hemerais tes sarkos autou, deeseis te kai hiketerias pros ton dunamenon (PPPMSA) sozein (PAN) auton ek thanatou meta krauges ischuras kai dakruon prosenegkas (AAPMSN) kai eisakoustheis (APPMSN) apo tes eulabeias:  (Heb 2:14; John 1:14; Romans 8:3; Galatians 4:4; 1 Timothy 3:16; 1 John 4:3; 2 John 1:7) (Psalms 22:1-21; 69:1; 88:1; Matthew 26:28-44; Mark 14:32-39; Leviticus 2:2; 4:4-14; John 17:1) (Matthew 27:46,50; Mark 15:34,37) (Isaiah 53:3,11; John 11:35) 

In the days of His flesh - The writer again emphasizes the reality of our Lord's humanity and His participation in all the experiences of His people. As Ironside says "He trod the path of faith and took the place of dependence on the Father."

He is now to show that Christ was under training for the priesthood, and describes the process of training.

Prayers (1162) (deesis) refers to urgent requests or supplications to meet a need and are exclusively addressed to God.  Deesis in the New Testament always carries the idea of genuine entreaty and supplication before God.  It implies a realization of need and a petition for its supply.

Supplications (2428) (hiketería form hiketes = a suppliant) originally described an olive branch entwined with wood carried by a suppliant. In the Greek culture the suppliant would hold and wave to express their desperate prayer and desire. The idea then came to mean that which is being urgently requested by someone, in this case the God-Man.

Lu 22:44 (being in agony He was praying very fervently) because of the anguish He faced in becoming sin for those who believed in Him. In the Garden of Gethsemane on the night before He went to the cross, Jesus prayed and agonized so intensely that He sweat great drops of blood. His heart was broken at the prospect of bearing sin.

Note that our Lord neither was saved from death nor did He ever pray to be saved from such. Furthermore He did He fear death as some teach. His mission as the Son of God was to enter our world and die. And yet  what a testimony these tears provide regarding the reality of His manhood!

Ironside recounts that...

Three times we read of His weeping. He wept at the grave of Lazarus as He contemplated the awful ravages that death had made, tears of loving sympathy. He wept as He looked upon Jerusalem and His prophetic soul saw the tribulations through which the devoted city must pass. And He wept in Gethsemane's garden as His holy soul shrank from drinking the cup of divine indignation against sin when He would hang on the cross. While the cup could not be averted, nevertheless He was heard because of His piety—that is, because of His godly fear, His reverence for the Father's will. And thus He who is the eternal Son who never knew what subjection meant, became man. As He walked the pilgrim path of suffering and rejection down here, He learned obedience by the things that He suffered. It is not that His will had to be subdued, but that from the moment when He assumed humanity He entered into new experiences. He who had always commanded learned practically what obedience meant. (Ironside Expository Commentary on Hebrews)

Crying (2906) (krauge from krazo = to croak or cry) is a cry which a man does not choose to utter but is wrung from him in the stress of some tremendous tension or searing pain. Krauge is  an onomatopoeic word, imitating the raven’s cry. It describes a crying, screaming, shrieking, shouting, Lat. clamor. The idea is a crying out as with a sharp cry or even a shout, as when one cries out in surprise (Mt 25:6), in support (Acts 23:9), in unrestrained brawling (Eph 4:31). In the present context this word pictures Jesus crying out loudly to God in prayer during His time of trial.

The rabbis wrote that

There are three kinds of prayers, each loftier than the preceding-prayer, crying and tears. Prayer is made in silence; crying with raised voice; but tears overcome all things.

TO THE ONE ABLE TO SAVE HIM FROM DEATH: pros ton dunamenon (PPPMSA) sozein (PAN) auton ek thanatou:  (Matthew 26:52,53; Mark 14:36) (Heb 13:20; Psalms 18:19,20; 22:21,24; 40:1-3; 69:13-16; Isaiah 49:8; John 11:42; John 17:4,5)

To the One able to save - Referring to God the Father.

Don't misunderstand what he is saying. Jesus was not hoping to escape the cross because it was for this very purpose that He came to earth. In John's gospel Jesus declared...

Now My soul has become troubled; and what shall I say, 'Father, save Me from this hour'? But for this purpose I came to this hour. (John 12:27)

Save Him from death - More literally this reads "save Him out of (ex) death". The point is that Jesus was not asking to be saved from dying but to be saved out of death or in other words to be saved from remaining in death. He was not asking to avoid the cross but to be assured of the resurrection (cf. Ps 16:8-11)

AND HE WAS HEARD BECAUSE OF HIS PIETY: kai eisakoustheis apo tes eulabeias: (Heb 12:28; Matthew 26:37,38; Mark 14:33,34; Luke 22:42-44; John 12:27,28)

and was heard in that he feared (KJV)

because of his reverent submission (NIV)

Guzik asks an interesting question...

 If Jesus asked that the cup be taken away from Him (Luke 22:42), and the cup was not taken away, how can it be said that He was heard? Because His prayer was not to escape His Father’s will, but to accept it - and that prayer was definitely heard.

Because of His piety - He was heard on account of His good acceptance of what He was accomplishing as the High Priest.

Piety (2124) (eulabeia from eulabes = careful as to the realization of the presence and claims of God, reverencing God, pious, devout from eu = good, well, right + lambano = take hold) in the original Greek usage meant caution, circumspection, discretion and then reverence or veneration. The Lxx usage in Joshua conveys the idea of fear, anxiety or dread. The NT uses convey the idea of godly fear, reverence, reverent regard, reverent submission or reverent awe in the presence of God.

Eusebeia is a closely related word is similar to eulabeia in reflecting an attitude of one's inner being, but in addition produces an demonstration of that inner  attitude in worship.

Vine writes that eulabeia...

signifies, first, “caution”; then, “reverence, godly fear,” Heb. 5:7; 12:28...in general, “apprehension, but especially holy fear,” “that mingled fear and love which, combined, constitute the piety of man toward God; the OT places its emphasis on the fear, the NT...on the love, though there was love in the fear of God’s saints then, as there must be fear in their love now” (Trench, Synonyms) (Vine, W E: Vine's Complete Expository Dictionary of Old and New Testament Words. 1996. Nelson)

There are only 2 uses of eulabeia in the NT...

Hebrews 5:7 (note) In the days of His flesh, He offered up both prayers and supplications with loud crying and tears to the One able to save Him from death, and He was heard because of His piety.

Hebrews 12:28 (note) Therefore, since we receive a kingdom which cannot be shaken, let us show gratitude, by which we may offer to God an acceptable service with reverence and awe

There are 2 uses in the Septuagint (LXX)...

Joshua 22:24 "But truly we have done this out of concern (eulabeia), for a reason, saying, 'In time to come your sons may say to our sons, "What have you to do with the LORD, the God of Israel?

Proverbs 28:14 How blessed is the man who fears (Hebrew = pachad - be in dread, in awe; LXX = eulabeia) always, But he who hardens his heart will fall into calamity.

The idea of eulabeia is that of being devoutly submissive. I wonder why my prayers don't seem to be heard so often? Could it be I lack this Philippians 2:5 attitude?

OUR BLESSÈD SAVIOR SEVEN TIMES SPOKE

Our blessèd Savior sev’n times spoke
When on the cross our sins He took
And died lest men should perish.
Let us His last and dying words
In our remembrance cherish.

“Father, forgive these men; for, lo,
They truly know not what they do.”
So far His love extended.
Forgive us, Lord, for we, too, have
Through ignorance offended.

Now to the contrite thief He cries:
“Thou, verily, in Paradise
Shall meet Me ere tomorrow.”
Lord, take us to Thy kingdom soon
Who linger here in sorrow.

To weeping Mary, standing by,
“Behold thy Son,” now hear Him cry;
To John, “Behold thy mother.”
Provide, O Lord, for those we leave;
Let each befriend the other.

The Savior’s fourth word was “I thirst.”
O mighty Prince of Life, Thy thirst
For us and our salvation
Is truly great; do help us, then,
That we escape damnation.

The fifth, “My God, My God, O why
Forsake Me?” Hark, the awful cry!
Lord, Thou wast here forsaken
That we might be received on high;
Let this hope not be shaken.

The sixth, when victory was won,
“’Tis finished!” for Thy work was done.
Grant, Lord, that, onward pressing,
We may the work Thou dost impose
Fulfill with Thine own blessing.

The last, as woe and sufferings end,
“O God, My Father, I commend
Into Thy hands My Spirit.”
Be this, dear Lord, my dying wish;
O heavenly Father, hear it.

Whoe’er, by sense of sin oppressed,
Upon these words his thoughts will rest,
He joy and hope obtaineth
And through God’s love and boundless grace
A peaceful conscience gaineth.

O Jesus Christ, Thou Crucified,
Who hast for our offenses died,
Grant that we e’er may ponder
Thy wounds, Thy cross, Thy bitter death,
Both here below and yonder.

ALTHOUGH HE WAS A SON HE LEARNED OBEDIENCE FROM THE THINGS WHICH HE SUFFERED: kaiper on (PAPMSN) huios emathen (3SAAI) aph on epathen (3SAAI) ten hupakoen aph on epathen (3SAAI) ten hupakoen:  (Hebrews 1:5,8; 3:6) (10:5-9; Isaiah 50:5,6; Matthew 3:15; John 4:34; 6:38; 15:10; Philippians 2:8)

One might say that Jesus' "training for the priesthood" involved suffering, even though he was the Son of God.

He learned obedience - The definite article ("the" in Greek) modifies obedience which identifies it as a specific or particular obedience which was required of Jesus in the days of His flesh.

The prophet Isaiah records these words of the Messiah...

The Lord GOD has opened My ear; And I was not disobedient, Nor did I turn back. I gave My back to those who strike Me, And My cheeks to those who pluck out the beard; I did not cover My face from humiliation and spitting. (Isaiah 50:5-6)

There is a principle that it is in the school of suffering where we grow the most in obedience. We all know only too well that often the best way to learn sympathy is by having suffered what another is suffering. We can read about the pain of starvation and even see pictures on television of starving children in Africa, but we until we have gone hungry for a period, we cannot completely sympathize with the victims of starvation. If suffering was fit to teach the Son of God, we must never despise it as a tool of instruction in our lives. If suffering was the lot of our Savior and Lord, His disciples are called to follow in His steps. Scripture does not teach that a dynamic faith will keep us from all suffering (more often a strong faith actually is associated with greater suffering!). The Scripture put suffering in an interesting light as the following representative passages illustrate...

(Paul reminds the believers at Thessalonica that) we sent Timothy, our brother and God's fellow worker in the gospel of Christ, to strengthen and encourage you as to your faith, 3 so that no man may be disturbed by these afflictions; for you yourselves know that we have been destined for this. (see note 1Thessalonians 2:2; 2:3)

And after they had preached the gospel to that city and had made many disciples, they returned to Lystra and to Iconium and to Antioch, 22 strengthening the souls of the disciples, encouraging them to continue in the faith, and saying, "Through many tribulations we must enter the kingdom of God." (Acts 14:21,22)

(Paul encourages the believers in Rome that they are) children, heirs also, heirs of God and fellow heirs with Christ, if indeed we suffer with Him in order that we may also be glorified with Him. 18 For I consider that the sufferings of this present time are not worthy to be compared with the glory that is to be revealed to us. (see notes Romans 8:17; 18)

Jesus suffered our sins as His own, and in this way He far more than any other priest. He is able to deal gently with our sins because He is so fully aware of the sense of personal defilement by sin, even though He Himself was sinless.

Henry Morris explains the obedience of our Lord this way...

What Jesus knew by omniscience, He "learned" by experience, thus "being made perfect"--not as God (for as God He was eternally perfect, by definition), but as man (Morris, Henry: Defenders Study Bible. World Publishing)

As Guzik says...

Jesus did not pass from disobedience to obedience. He learned obedience by actually obeying. Jesus did not learn how to obey; He learned what is involved in obedience. (Hebrews 5)

In short, Jesus learned the full meaning of the cost of obedience from the things which He suffered, and God therefore affirmed Him as the completely obedient, perfect High Priest. That is the kind of High Priest weak men need, One Who knows and understands what we are going through.

A modern poet says of the poets

“We learned in suffering what we teach in song.”

William Barclay make a good point writing that...

God speaks to men in many experiences of life, and not least in those which try their hearts and souls. But we can hear his voice only when we accept in reverence what comes to us. If we accept it with resentment, the rebellious cries of our own heart make us deaf to the voice of God. (Barclay, W: The Daily Study Bible Series. The Westminster Press or Logos)

The pardon of a believer's sins is an entire pardon. It is the full pardon of all his sins. It was no pardon to him if it were not an entire pardon. If it were but a partial blotting out of the thick cloud-if it were but a partial canceling of the bond-if it were but a forgiveness of some sins only, then the gospel were no glad tidings to his soul. The law of God had brought him in guilty of an entire violation. The justice of God demands a satisfaction equal to the enormity of the sins committed, and of the guilt incurred. The Holy Spirit has convinced him of his utter helplessness, his entire bankruptcy. What rapture would kindle in his bosom at the announcement of a partial atonement-of a half Savior-of a part payment of the debt? Not one throb of joyous sensation would it produce. On the contrary, this very mockery of his woe would but deepen the anguish of his spirit. But go to the soul, weary and heavy-laden with sin, mourning over its vileness, its helplessness, and proclaim the Gospel. Tell him that the atonement which Jesus offered on Calvary was a full satisfaction for his sins;-that all his sins were borne and blotted out in that awful moment;-that the bond which Divine justice held against the sinner was fully cancelled by the obedience and sufferings of Christ, and that, appeased and satisfied, God was "ready to pardon." How beautiful will be the feet that convey to him tidings so transporting as this! And are not these statements perfectly accordant with the declarations of God's own word? Let us ascertain. What was the ark symbolical of, alluded to by the apostle, in the ninth chapter of his Epistle to the Hebrews, which contained the manna, Aaron's rod, and the tables of the covenant, over which stood the cherubim of glory shadowing the mercy-seat? What, but the entire covering of sin? For, as the covering of the ark did hide the law and testimony, so did the Lord Jesus Christ hide the sins of His chosen, covenant people-not from the eye of God's omniscience, but from the eye of the law. They stand legally acquitted. So entire was the work of Jesus, so infinite and satisfactory His obedience, the law of God pronounces them acquitted, and can never bring them into condemnation. "There is therefore now no condemnation to those who are in Christ Jesus; who walk not after the flesh, but after the Spirit." "Who is he that condemns? It is Christ that died, yes rather, that is risen again who is even at the right hand of God, who also makes intercession for us."