The Aruch Ha'shulhan (Rav Yechiel Michel Epstein, 1829-1908) writes that the text of the Kaddish prayer was likely written by the Ansheh Kenesset Ha'gedola ("Men of the Great Assembly") during the first years of the Second Commonwealth. This prayer expresses the wish that G-d's Name should be glorified and become known throughout the world. The destruction of the first Bet Ha'mikdash marked a grave Hilul Hashem – desecration of G-d's Name – and so when Jews returned to their land and began rebuilding the Temple, the Rabbis composed this special prayer that the glory of G-d's Name should be restored.
The Yalkut Yosef (Hebrew edition) notes that this theory might be supported by the Gemara's comment (Berachot 33a) that our prayers and blessings were written by the Ansheh Kenesset Ha'gedola. However, as noted by Rav Yisrael Bitan, the Gemara did not specifically mention Kaddish, and thus this proof is not conclusive.
In any event, this is the opinion followed also by Rav Shlomo of Worms (Germany, d. 1096), in his Siddur. He explains that after seventy years in Babylonian exile, the Jews' primary language was Aramaic, instead of Hebrew, and for this reason the Kaddish text was written in Aramaic.
A different view is presented by the Orhot Haim (Rav Aharon of Lunel, late 13th-early 14th century), who maintained that the Kaddish text was written several generations later, by the Tanna'im.
The unique significance of the Kaddish prayer, and its precious value, is clearly expressed in several passages in the Gemara.
In Masechet Berachot (3a), the Gemara tells that Rabbi Yossi was once traveling and stopped to pray in one of the ruins of Jerusalem. While he was there, he heard a voice weeping and lamenting, "Woe unto the children because of whose iniquities I destroyed My home, burned My sanctuary, and exiled them among the nations." Afterward, he was informed by Eliyahu the Prophet that this cry is sounded three times each day. However, Eliyahu added, when Jews assemble in synagogues and study halls and pronounce in Kaddish, "Yeheh Shemeh," the Almighty "nods His head," so-to-speak, and regrets having driven the Jewish People into exile. The Kaddish recitation thus arouses G-d's love and compassion, and brings the final redemption closer.
Moreover, the Gemara teaches in Masechet Shabbat (119b) that if one answers "Yeheh Shemeh Rabba" with all his might, harsh decrees that were issued against him are rescinded. And the Gemara states in Masechet Sota (49a) that since the destruction of the Bet Ha'mikdash, the world's condition has been increasingly worsening, yet the world survives in the merit of "Kiddusha De'sidra" and the Kaddish recited after Torah study. ("Kiddusha De'sidra" refers to the section known to us as "U'ba Le'sion," when we cite several verses followed by their Aramaic translation.)
Rav Amram Gaon (9th century) tells that Rabbi Yishmael was once shown by an angel the horrific tragedies that were decreed to befall the Jewish People. The angel explained that new decrees are issued against the Jews every day, but these decrees are left unfulfilled in the merit of the Jews' recitation of "Yeheh Shemeh Rabba."
There was once a member of our community who suffered a stroke, and the family, who heard of the great power of Kaddish to annul harsh decrees, brought a Minyan to the rehabilitation center. They prayed there with the patient, ensuring to have special Kavana (concentration) when responding "Yeheh Shemeh Rabba." The patient quickly recovered, returned to work, and lived for many years – a clear demonstration of the special power of Kaddish.