I should add that this cry of Jesus is really a hard piece of work. On the one hand, this seems to be an ‘embarrassing’ / ‘against-the-grain’ (whichever term you prefer) tradition, but on the other hand, the narrative itself at this point is so drenched with Scriptural allusions that one could be tempted to ask whether the cry could be a ‘historicization’ of Scripture. (If we choose here to follow John Dominic Crossan’s idea of the passion narratives being ‘prophecy historicized’, that is.

)
15:23 And they offered him
wine mixed with myrrh, but he did not take it. (= Psalm 69 (68):22 LXX “And they gave gall as my food, and for my thirst they gave me vinegar to drink.” Note that Matthew has a more overt parallel at this point.)
15:24 And they crucified him and
divided his garments among them, casting lots for them, to decide what each should take. (= Psalm 22 (21):19 “They divided my clothes among themselves, and for my clothing they cast lots.”)
15:27 And with him they crucified two robbers, one on his right and one on his left. (= Isaiah 53:12 “And he was reckoned among the lawless.” A particular textual variant (the missing verse 28) actually quotes this passage perhaps under influence from Luke 22:37.)
15:29-30 And those who passed by
derided him, wagging their heads… (Psalm 22 (21):8; 109:25 “All who saw me mocked at me; they talked with the lips; they moved the head;” “And I, I became a reproach to them; they saw me; they shook their heads.” Also cf. Wisdom of Solomon 2:12-20 for the mockery.)
15:33 And when the sixth hour had come, there was darkness over the whole land until the ninth hour. (= cf. Amos 8:9-10.)
15:36 And someone ran and filled a sponge with
vinegar, put it on a reed
and gave it to him to drink, saying, “Wait, let us see whether Elijah will come to take him down.” (= Psalm 69 (68):22 again.)
15:40 There were also women looking on
from a distance, among whom were Mary Magdalene, and Mary the mother of James the younger and of Joses, and Salome. (Greek
apo makrothen; Psalm 38:11 LXX “My friends and my fellows approached opposite me and stood, and my next of kin stood
afar off (
apo makrothen).”)
The above text is actually a butchered, partial translation of Psalm 22:1. The word, ‘eloi is neither Hebrew nor Aramaic. The psalm reads, ‘eli while another form of “my god” in Hebrew is 'elohi with the Aramaic equivalent being 'alahi. It appears that Jesus failed to remember the correct Hebrew form of “My God.” He tried but failed to quote accurately. This is why the remainder of the quote is in good Aramaic, his native vernacular. Jesus knew that he misspoke the Hebrew word for “My God” and so abandoned the attempt and resorted to a translation.
The form that Jesus came up with moments before he died not only proves to be embarrassing, but also satisfies another criterion used to isolate the authentic words of Jesus–Orality. If we seek to account for the unattested form, 'eloi, the conclusion is at hand that Jesus mistakenly conflated two Hebrew words, 'eli, which is the form in Ps 22:1, and 'elohi, a common form in Hebrew for “my god.” Accordingly, Jesus took the holem (long o vowel in Hebrew) and unconsciously inserted it into 'eli, thereby producing and indeed inventing the form as we have it in Mark. Thus, it appears that we have here an oral expression since it is highly unlikely that some scribe would misquote Ps: 22:1 who would have had the correct form before him. That this is true is clinched by the fact that Matthew, working form Mark’s written form, corrects Mark to the Psalm. Matthew was embarrassed by Mark’s form. Therefore, these words satisfy two of the criteria for isolating the words of Jesus.
I think you’re reading too much into the transliteration
Elōi. You’re correct that a number of people argue that the transliteration of the Aramaic
'elāhī into Greek is a bit off, but IMHO at this stage we could never be sure whether this is Jesus’ fault or Mark’s. Assuming that Matthew did change Mark’s transliteration I personally don’t think this is so much out of ‘embarassment’ (where is the embarassment?) but to draw a more clearer parallel to ‘Elijah’.
As an aside, I should mention here the fact that the different manuscripts themselves could not agree on how to spell these words: whether it was
ēli - as in Matthew - or
elōi,
lama or
lema or
lima, and
sabachthani or
sabaphthani or
sabaktani or
zaphthani. Also, there’s a particularly interesting quote from Judges 5:5 (version B) in the Septuagint: ὄρη ἐσαλεύθησαν ἀπὸ προσώπου
κυρίου Ελωι τοῦτο Σινα ἀπὸ προσώπου κυρίου θεοῦ Ισραηλ. “Mountains quaked from before
the Lord Elōi, this Sinai, from before the Lord, God of Israel.” The word isn’t found in the Hebrew version (or in Greek version A), but this is a rather curious instance of
Elōi.