"Eli Eli lama sabachthani?" is Greek transliteration of Aramaic words. If it was Hebrew, then azabthani would have been used instead of Aramaic word "Sabachthani."
Check this link for Hebrew NT of Matthew 27.
In Hebrew, "Eli Eli lama sabachthani?" will become "Eliy ‘Eliy lamah `azab’taniy?"
Here is an explanation I found on why people thought Jesus was calling for Elijah.
"Jesus Christ was suffering horrendous pain for about SIX HOURS. He called out for “Eli”, His exhaustion and heavy breathing could have caused Him to add an “ah” on the end. Try talking when you have gone for a long run and you’ll see what I mean. “Eli-ah” sounds a lot like “Eliyah” does it not?" (Source - http://ellhn.e-e-e.gr/books/assets/NewTestament.pdf on Page. 52)
Eliya (also written as Eliyah in English) is Aramaic for "Elijah." (Source - Mark 15:35 of Aramaic Peshitta New Testament)
I also want to point out that the spoken language of first century Israel was Aramaic. Not Hebrew or Greek. This is confirmed by New Testament and Jewish Historian Josephus.
Jewish Historian Josephus states that Jews didn't speak Greek in first century Israel.
Jewish Historian Josephus wrote:
"I have also taken a great deal of pains to obtain the learning of the Greeks, and understand the elements of the Greek language, although I have so long accustomed myself to speak our own tongue, that I cannot pronounce Greek with sufficient exactness; for our nation does not encourage those that learn the languages of many nations, and so adorn their discourses with the smoothness of their periods; because they look upon this sort of accomplishment as common, not only to all sorts of free-men, but to as many of the servants as please to learn them. But they give him the testimony of being a wise man who is fully acquainted with our laws, and is able to interpret their meaning; on which account, as there have been many who have done their endeavors with great patience to obtain this learning, there have yet hardly been so many as two or three that have succeeded therein, who were immediately well rewarded for their pains." - Antiquities of Jews XX, XI
Jewish Wars (Book 1, Preface, Paragraph 1) -
"I have proposed to myself, for the sake of such as live under the government of the Romans, to translate those books into the Greek tongue, which I formerly composed in the language of our country, and sent to the Upper Barbarians. Joseph, the son of Matthias, by birth a Hebrew, a priest also, and one who at first fought against the Romans myself, and was forced to be present at what was done afterwards, [am the author of this work]."
Also note the names in English Bible of New Testament - "Bar"tholomew, "Bar"abbas, "Bar"nabbas, "Bar"sabbas, "Bar" Jesus, Simon "Bar" Jonah, "Bar" Timaeus, etc.
Aramaic word Bar means Son. In Hebrew, Ben means Son. For example, Benjamin in Old Testament, and First Israeli Prime Minister, David Ben Gurion.
"And it became known to all the inhabitants of Jerusalem, so that the field was called in their own language Akeldama, that is, Field of Blood."
"Akel dama" is Greek transliteration of Aramaic words "Khqel Dama." "Khqel Dama" is also written as Hakel Dama in English.
We clearly see "Field of Blood" was called "Khqel Dama" by all the inhabitants of Jerusalem in their own language which is Aramaic.
If I translate aramaic words "Khqel Dama" into Hebrew, then "Khqel Dama" will become "Sh'deh Hadam."
Through this, we can read that all inhabitants of Jerusalem spoke in their own language in first century AD which was Aramaic. If Hebrew was used as spoken language in first century Israel, then "Sh'deh Hadam" would have been mentioned along with "Khqel Dama" (a.k.a akel dama in Greek and English NT) in Acts 1:19.
Here is the link to Acts Chapter 1 (Hebrew translation from Greek)
You will see "s'deh Hadam" at the end of Acts 1:19. To match the words, see S'deh (Green color) and Field (Green Color). Hadam (in purple color) and Blood (in purple color).
Josephus' Antiquities of Jews 3:32 - Now the Hebrews call this food manna; for the particle man, in our language, is the asking of a question, What is this?
"Man" is Aramaic. In Hebrew, Aramaic word "Man" will become "Mah".
According to Dead Sea Scrolls archaeologist, Yigael Yadin, Aramaic was the language of Jews until Simon bar Kokhba tried to revive Hebrew and make Hebrew as the official language of Jews during Bar Kokhba revolt (132-135 AD). Yigael Yadin noticed the shift from Aramaic to Hebrew during the time of Bar Kokhba revolt (132 - 135 AD). In Yigael Yadin's book "Bar Kokhba: The rediscovery of the legendary hero of the last Jewish Revolt Against Imperial Rome", Yadin notes,
"It is interesting that the earlier documents are written in Aramaic while the later ones are in Hebrew. Possibly the change was made by a special decree of Bar-Kokhba who wanted to restore Hebrew as the official language of the state" (page 181).
In Book "A Roadmap to the Heavens: An Anthropological Study of Hegemony among Priests, Sages, and Laymen (Judaism and Jewish Life)" by Sigalit Ben-Zion (Page 155), Yadin remarked: "it seems that this change came as a result of the order that was given by Bar Kokhba, who wanted to revive the Hebrew language and make it the official language of the state."
According to Book "Naming the Witch: Magic, Ideology, and Stereotype in the Ancient World" written by Kimberly B. Stratton (p. 232), Yadin suggests that Bar Kokhba was trying to revive Hebrew by decree as part of his messianic ideology.