In all Christian Bibles the translators wrote this verse in the same language and translated it, (maybe it's Hebrew which I don't know) but why only Mark 15:34 in the whole, is it because when Jesus was crying He cried in another language?
It's Syriac. Matthew Henry's commentary in the sidebar at this link says
The sentence is a direct quotation of Psalm 22:1, and as well as Mark 15:34 it does appear in the same form at Matt 27:46. Matthew Henry's commentary on the Matthew verse in the sidebar at this link says
The New Testament was written in Greek, but the Greek text records Jesus' words in Aramaic (in Mark, Hebrew in Matthew). The Gospel writers transliterated the Aramaic (Mk 15) and Hebrew (Mt 27) into the Greek script.
It is important here to distinguish between script and language. For instance, I can write in Spanish, Latin, German, English, etc. all with (basically) the same Latin script, even though the languages are all different. And likewise, I can take the Hebrew word אֶל and transliterate it into Latin characters, eʾl. The language is still Hebrew, but the script is Latin.
Short answer: It's Aramaic (or Hebrew in Matthew) transliterated into the Greek script. Since the text provides a Greek translation immediately thereafter, it makes sense that the translators would retain it.
The ancient Eastern text translation, "My God, My God, for this (cause/purpose) was I spared," agrees with Jesus' own declaration to His disciples in John 12:27:
In the following verse, Jesus asks, "Father, glorify Thy name. Then came there a voice from heaven, saying, I have both glorified it, and will glorify it again."
When those standing nearby heard God answer Jesus, they couldn't agree on what they heard. Jesus, in answer, replied in the following verse.
These verses from John, taken together with the cry of Jesus to the unbelieving masses in Matthew 27:46 and Mark 15:34 as translated in the ancient Eastern text, fit hand in glove with Christ's purpose on earth.
I concur with Waeshael.
Let me answer this question from the perspective of an Eastern Christian Tradition where Aramaic NT is used instead of Greek NT.
Eloi, Eloi, lama sabachthani is a clumsy Greek transliteration of Aramaic words "Elahi, Elahi, lmana shwaqthani?"
Greeks often clumsily transliterates Aramaic words. For Example, Haqal Dama is transliterated as Akel Dama in Greek NT of Acts 1:19 (Source - Book "What do Jewish People think about Jesus?" by Dr. Michael Brown, Page 39).
In Aramaic, Eil and Elaha are the words for God. Hebrews in Judea commonly preferred "Eil" while "Elaha" is commonly preferred in Galilee and Syria.
For Example, Americans prefer to use the word "Yuca or Cassava" while People in England prefer to use the word "Tapioca."
So "Eil Eil Lmana Shwaqthani?" in Matthew 27:46 of Peshitta (Source - http://www.peshitta.org/pdf/Mattich27.pdf) is transliterated as "Elahi, Elahi, lmana shwaqthani?" by Mark 15:34 (Source - http://www.peshitta.org/pdf/Marqsch15.pdf)
Aramaic is also known as Syriac, because Greeks called Arameans or Aramites as Syrians or Syriacs. So Aramaic came to be known as Syriac among Greeks.
Poseidonios from Apamea (ca. 135 BC - 51 BC) - "The people we Greek call Syriacs, they call themselves Arameans." [Source - J.G. Kidd, Posidonius (Cambridge Classical Texts and Commentaries, 1988), vol. 2, pt. 2, pp. 955-956].
Josephus wrote - "Aram had the Aramites, which the Greeks called Syrians."(Source - Antiquities of the Jews, translated by William Whiston in 1737, Book I, Chapt. 6, Paragraph 4).
German Orientalist Theodore Nöldeke also wrote this:
"It is well understandable that people have started to transfer the name of the country to the most important nationality and so the name 'syriac' was apprehended ethnological and was equated with 'aramaic'." [Source - Theodor Nöldeke: Book "Assyrios Syrios Syros, in Zeitschrift für klassische Philologie", Hermes 5, Berlin 1871, p. 461)].
In first century AD, Josephus points out that Aramaic was a widespread language and understood accurately among Aramaic speakers.
Josephus points out how Parthians, Babylonians, the remotest Arabians, and those of his nation beyond Euphrates with the Adiabeni knew accurately about Jewish Wars by his means (mentioned below) and why he translated his work from the language of his country into Greek for Greeks and Romans.
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]."
Jewish Wars Book 1 Preface, Paragraph 2 - "I thought it therefore an absurd thing to see the truth falsified in affairs of such great consequence, and to take no notice of it; but to suffer those Greeks and Romans that were not in the wars to be ignorant of these things, and to read either flatteries or fictions, while the Parthians, and the Babylonians, and the remotest Arabians, and those of our nation beyond Euphrates, with the Adiabeni, by my means, knew accurately both whence the war begun, what miseries it brought upon us, and after what manner it ended."
In first century AD, Greek wasn't spoken in first century Israel. This is confirmed by the testimony of Jewish Priest Josephus who points out that Greek wasn't spoken in first century Israel and the extreme rarity in terms of a Jew knowing Greek.
"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.
More information about this can be read here (with sources) - http://en.metapedia.org/wiki/New_Testament
The unity of Aramaic (a.k.a Syriac) among Aramaic speakers began to fall apart only after Simon Bar Kokhba came to power in 131/132 AD. It was Simon Bar Kokhba who revived Hebrew and tried to make Hebrew as the official language of the state during Bar Kokhba revolt (132-135 AD) [Source - 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)].
More information about this can be read here - http://en.metapedia.org/wiki/Language_of_Jesus
The expression is Aramaic - a Syrian dialect. The Greek author did not understand the expression. Luckily the English translation retained the underlying Aramaic sentence structure. Sometime during the war with the Turks in the first world war, some isolated villages were discovered where the locals spoke Aramaic, and used an Aramaic bible. Scholars from Oxbridge were sent to teach the English language, and so the first Aramaic/English lexicon was born. By 1938 it was published, and for the first time we were able to understand, not only the meaning of the words but also the meaning of the many colloquialisms. In the Aramaic bible of the East, in the introductory notes by the translator, who was an Aramaic speaker (Lamsa) he explains how easy it was for the Greeks to misread the small inflections of the Aramaic symbols, and so mistranslate the words. The phrase mentioned, would be better translated as "My God, My God, for this was I born." It seems that the Greek translator selected from the OT a phrase appropriate to the situation. In those days there was no agreement as to whether or not God was on the cross, influential Bishops thought not, so the expression was translated to confirm the humanity of Jesus over the gnostic teaching of some Bishops that Jesus was divine himself and therefore could not have suffered. This was a heated argument for a century or so, and was not fully decided until the first ecumenical council in the 4th. Cent. The translation of this phrase in our bible is western catholic tradition. So much commentary has been published on this phrase that it seems a bit unkind to make any change to text.