Tosafot (Berachot 3a) addresses this question.

1. The first explanation that Tosafot brings is what he refers to as the "common explanation" for this variation in the liturgy. It is claimed, says Tosafot, that the Kaddish prayer was written in Aramaic due to its unusual holiness. The Kaddish is such an eloquent testimony to Hashem's divine glory, that were we to recite it in Hebrew, it might arouse the jealousy of the angels in heaven! We therefore say this praise in Aramaic, a language that, according to the Talmud (Shabbat 12b), the heavenly hosts do not "recognize." (The angels do not heed what is said in Aramaic, since Aramaic was not usually reserved for prayer but for mundane, everyday conversation -- see Tosafot Shantz to Sota 33a; T'mim De'im #184 in the name of Rav Yehudah bar Yakar.)

Tosefot, however, does not accept this explanation. We have plenty of beautiful prayers, and they all are in Hebrew! Why should this prayer arouse the jealousy of the angels more than any other?

2. Tosafot therefore suggests an alternate solution for the Kaddish mystery. Kaddish was recited not only during the daily prayers, but also at the conclusion of the public sermons that were normally delivered on Shabbat. The broad audience that attended these sermons included many unlearned folk who were not accustomed to the Hebrew language (i.e. they didn't even pray regularly. It is interesting to note that until today, people who do not pray regularly still come to the synagogue to recite the Kaddish prayer). It was for the benefit of these people that Kaddish was composed in the vernacular, rather than in the Holy Language that was usually reserved for such matters.

3. T'mim De'im (ibid.) offers another suggestion. The Gemara (Shabbat 12b) tells us that we should only pray in Hebrew (as opposed to Aramaic, or the vernacular) in order that the angels of heaven, who don't recognize Aramaic, should be able to bring our prayers before the Almighty. Without their help, it is apparently much harder for our prayers to accomplish their goals. (A more in depth discussion of this point will have to wait for another occasion.) The Gemara (Sota 33a), however, presents an exception to this rule. Prayers offered by a group (of ten or more Jewish men) are immediately accepted even without the intervention of the heavenly hosts.

This, suggests the T'mim De'im, is why the Kaddish was written in Aramaic. All prayers that are sometimes prayed alone, were instituted in Hebrew, the language that the angels recognize, so that the angels can be involved. Prayers that are always prayed in groups, and are not appropriate for the individual for one reason or another, were composed in Aramaic as well. The Kaddish prayer, although recited by an individual, is always said in the presence of a group. Therefore, it is considered to be a "group prayer," and may be recited in Aramaic.

Is there any way to defend the "common explanation" (1) that Tosafot quotes? Why should the angels get more upset over Kaddish than over other beautiful prayers? A beautiful answer to this question is presented by the book "Ve'Imru Amen" (Rav Yehoshua Alter Wildman, Jerusalem 1929, vol. II #59 -- pp.90-91), in the name of Rav Menachem Mordechai Frenkel of Jerusalem.

Yakov called his children and said, "Gather around, I wish to tell you what will befall you in the end of days!"

(Bereishit 49:1)

Yakov was about to reveal to his children the time when Hashem would redeem their descendants from exile [with the coming of the Moshiach], when the Divine Presence suddenly withdrew from him. "Perhaps one of my children isn't worthy?" wondered Yakov. His children answered him, "Shema Yisroel Hashem Elokeinu Hashem Echad" -- "Hear, our father Yisrael, the Lord is our G-d; the Lord is one!" Our hearts are united in the service of Hashem. At that point, Yakov exclaimed, "Baruch Shem Kevod Malchuto Le'olam Va'ed" -- "Blessed be the Name of His glorious kingdom for all eternity!"

(Gemara Pesachim 56a)

The Targum Yerushalmi (a verse by verse Aramaic translation and Midrashic commentary on the Torah) to Bereishit 49:1, brings a slightly different version of the Aggadah that we quoted above from Pesachim. According to the Targum, instead of proclaiming "Baruch Shem...," Yakov declared "Yehe Shemei Rabbah..." -- the phrase from the Kaddish prayer!

This discrepancy, however, can be easily explained. The Gemara in Pesachim (which tells the story in Hebrew) quotes the Hebrew version of Yakov's reply. This is "Baruch Shem Kevod Malchuto Le'olam Va'ed." The Targum, though, tells the story in Aramaic. Perhaps "Yehe Shemei Rabbah ..." (May His great Name be blessed, now and for all eternity!) is simply an Aramaic rendering of the Hebrew "Baruch Shem..." (Blessed be the Name of His glorious kingdom for all eternity!). The slight difference between the two is acceptable. Targum [= an Aramaic rendering] is not meant to be a *literal* translation, but rather it is meant to convey the idea contained in the Hebrew phrase (Gemara Kiddushin 49a -- see Tosefot ad loc.).

"When Moshe was in heaven (to receive the Torah), he heard the angels praising Hashem, "Baruch Shem... ." When he returned, he taught this praise to the Bnai Yisroel. Why don't we say this praise out loud? Rav Assi said, "It can be compared to a man who stole jewelry from the king's palace. He gave it to his wife and told her, "Don't wear this in public, so that the king should not find out that you have it. Wear it only in the house!"

(Devarim Rabba 2:25, quoted by Tur Orach Chaim #619)