Most importantly, we have to ask what it is that Christ has specifically done “in the days of his flesh.” First, those actions are put forward with the continuing theme in mind of testing and passing the test. Christ “offered up prayers and supplications with loud cries and tears to the One [God] able to deliver him out of death, and he was heard because of his piety.” Even the Son of God felt apprehension and prayed for deliverance, and because of his faith in God was indeed delivered—that is, he was resurrected, not spared his suffering and death.
Not that it is of any great importance in this case, since context is queen, but Doherty does not actually argue for his reading of salvation out of death (ἐκ θανάτου) as salvation after death; he simply assumes that some sort of resurrection is in view. Now, I think he is correct in this, but I also want to present a relevant parallel to it, since it turns out that the expression usually seems to entail, at least in the LXX, an evasion of death (that is, the saved one does not die and get resurrected, but rather is saved from having to die). There are numerous examples of this meaning, only one of which I will actually cite verbally, to wit, Esther 4.8b LXX:
...and speak to the king on our behalf and rescue us out of death [ἐκ θανάτου].
Other instances are found in Joshua 2.13; Psalm 33.18-19 (32.18-19 LXX); Psalm 56.13 (55.14 LXX); 116.8 (114.8 LXX); Proverbs 10.2; 23.14; Job 5.20; 33.30; Ezekiel 12.16; Hosea 13.14; epistle of Jeremiah 1.35; and Tobit 12.9. This usage is quite common.
But there is one instance in which the phrase ἐκ θανάτου clearly refers to resurrection, and that is Sirach 48.5a, which is directly addressing the prophet Elijah:
You who raised a corpse out of death [ἐκ θανάτου] and from Hades....
That is the usage we have on our hands in Hebrews 5.7, I think. Even though the phrase usually suggests an evasion of death, it does not have to do so; it can be used of a resurrection from the dead, as well.
Doherty uses this point to suggest that Gethsemane is not in view in this epistolary passage. Basically, in the synoptic gospels (Matthew 26.36-46 = Mark 14.32-42 = Luke 22.39-46), Jesus prays that the cup might pass; in other words, he prays not to have to suffer death at all. In Hebrews 5.7-8, however, the son prays to be saved out of death, that is (in this case), resurrected, and his prayer is heard (εἰσακουσθεὶς), suggesting that he gets what he prays for. Therefore, he is not praying to escape death, as he is in the synoptics.
I might add that the praying that Jesus does in this epistolary passage is not necessarily to be confined to a single event, anyway, since it is introduced as something that happened in the days (plural) of his flesh. This is obviously not a lock, since it is not a non sequitur to speak of a single event happening during a stretch of time, but it does leave open the distinct possibility that more is in mind than one stressful night.
Doherty also rightly cites scholars who point to the possible derivation of Hebrews 5.7-8 from scriptural precedents. His scholars point to two verses from the Psalms that employ descriptive words from Hebrews 5.7. Psalm 22.24b (21.25b LXX) affirms that in my crying toward him he heard me (καὶ ἐν τῷ κεκραγέναι με πρὸς αὐτὸν εἰσήκουσέν μου), and Psalm 116.1 (114.1 LXX) says that the Lord will hear the voice of my request (εἰσακούσεται... τῆς φωνῆς τῆς δεήσεώς μου). I would add that the specific combination of requests and supplications appears also in Job 41.3 (40.27 LXX), though admittedly in a context unlikely to serve as the actual inspiration for the passage:
But will he [that is, leviathan] speak softly to you with request and supplication [λαλήσει δέ σοι δεήσει ἱκετηρίᾳ μαλακῶς]?
At any rate, I find no necessary connection to Gethsemane in this passage. (It is, in fact, not beyond the realm of possibility that this passage, or something like it, contributed to the Gethsemane account.)
Finally, I would like to present my own rendition of the full extent of Hebrews 5.5-8, since in the Greek it really all comes out as one long sentence held together by subordinate clauses (for convenience I have underlined a relative pronoun and the verb that fulfills it only after the interjection of several participial phrases):
So also Christ did not glorify himself to become a high priest, but (rather it was) he who said to him: You are my son; today I have begotten you; just as he says also in another passage: You are a priest forever according to the order of Melchizedek, (Jesus) who, in the days of his flesh having offered forth both requests and supplications with loud crying and tears to the one able to save him out of death, and having been heard because of his piety, although being a son [καίπερ ὢν υἱὸς], learned obedience from the things which he suffered.
This version bucks the trend of the usual translations in that it links the participial phrase although being a son with what goes before rather than what comes after it. The sense would be that Jesus was heard because of his piety, even though he was a son; in other words, his prayer life was not cut any nepotistic slack; he relied on his pietistic standing with God just like everybody else (with the difference that he was actually sinless; see Hebrews 4.15).
Taking the participial phrase set off by καίπερ (although) with what goes before finds precedent elsewhere in the epistle. Take Hebrews 7.5, for example:
And those from the sons of Levi who receive the priesthood have a command according to the law to exact a tithe from the people, that is, from their brethren, though they have come out [καίπερ ἐξεληλυθότας] of Abraham.
Hebrews 12.17 works the same way. I am not saying that the participial phrase cannot go with the next clause, but I think the verse makes more sense if it goes with the preceding clause.
Such are my thoughts so far on these verses.
Ben.