Jesus cries out using the words of Psalm 22, and the people standing at the foot of the cross say He’s calling for Elijah.  Why?  They must be Jews, since they immediately associate “Eli” with “Elijah.”  But if they are Jews, why don’t they recognize a quotation from one of the Psalms?  Have they become so dull of hearing?  Perhaps that’s Matthew’s point, since the Jews have clearly been incapable of noticing the fulfillments of Scripture that strew the ground all around Jesus’ cross - the very wording of the mockery, gambling for clothes, vinegar to drink, all of it is derived from the Scriptures but the Jews don’t recognize it.

Maybe.  It’s a bit easier, though not easy, to determine why Matthew would have included the speculation concerning Elijah.

Elijah of course ascended to heaven without dying; he was not crucified.  But the original Elijah narrowly escaped death.  Ahab was looking for him everywhere (1 Kings 18:7-15), and especially after he slaughtered the prophets of Baal Jezebel threated to kill him (1 Kings 19:2).  He fled into the wilderness, a single Israel spending 40 days and 40 nights fasting in the desert until he arrived at Horeb (1 Kings 19:8).  At Horeb, he complained to Yahweh that He had done nothing about the attacks on the covenant, altars, and prophets; he complained that he alone was left, forsaken by all, and in danger of death (1 Kings 19:9-10, 13-14).  In response to Elijah’s complaint, Yahweh passed by, preceded by a strong wind, the tearing of mountains, the breaking of rocks, and a fire, finally coming near in a voice (1 Kings 19:11-12).

The match with the crucifixion is not extremely tight, but it’s tantalizing.  Jesus has already been mistaken for Elijah (Matthew 16:14) and compared His approaching death to the death of the new Elijah, John (Matthew 17:10-12).  Elijah ascended to heaven; the new Elijahs, both John and Jesus, are put to death.  Jesus is isolated and alone; “I alone am left” describes Jesus much more accurately than Elijah.  Jesus cries out in complaint to His Father.  Jesus cries out twice, as Elijah presents his complaint before Yahweh twice.  In response to the first complaint of Elijah, Yahweh brings wind, split rocks, earthquake, and fire; in response to the second cry of Jesus, the Father splits the veil of the temple, sends an earthquake, splits rocks, and opens tombs.  1 Kings 19 and Matthew 27 are two of the three places I’ve found that speak of the Lord breaking or splitting rocks (the other is Psalm 78, a reference to the Lord splitting the rock in the wilderness to give water).

Thus, on the cross Jesus cries out before Yahweh.  But the cross is also the mountaintop from which Jesus, another Elijah, brings the covenant lawsuit against His people, and is assured that His Father is turning the world upside down in response.