The Maharal (Derech Chayim, 5, 23) writes that the notion of Aramaic being the common vernacular at that time is tenable as to why the Talmud Bavli was written in Aramaic, however with regard to the Talmid Yeushalmi, where the primary language was Hebrew, such an answer is unsatisfactory. On a much greater scale, we find Aramaic in the Torah itself (Bereishis, 31, 47) "יגר שהדותא" and in Nach (Yirmiyahu, 10, 11) "כדנה תימרון להון" and in a large part of the Book Daniel (See Sotah 30a).
In order to answer this question, we have to ask ourselves an even bigger question as to what Aramaic is in general and what is its relationship to Torah. That is a much more complex question and is beyond the scope of this post. I will list sources of reference, however, so you can learn more about the subject.
Maharal: Derech Chayim - 5, 22, 23.
Chiddushei Aggados - Sanhedrin (38b) [This is the most fundamental source]
Tiferes Yisrael - 13.
Gevuros Hashem - 54.
Rav Tzadok Hakohen Milublin: Kometz Haminchah - Part 2, #79.
Rav Yosef Gikatilia: Shaarei Orah - 10.
Rav Nasson Nata Shapira: Megaleh Amukos - Parshas Vayeitzieh and Bahaloscha.
Rav Yonason Eibishitz: Yaaros Devash - Drush 1.
Rav Nachman Mibreslov: Likkutei Moharan - Mahadura Kamam, 19.
Contemporary Rabbonim have discussed this idea at length clearly like Rav Reuven Chayim Klein Shlit"a in his sefer Lashon Hakodesh.
The main work that I find to be the clearest
and most comprehensive is the sefer Ksav Ivri - Ksav Ashuri (pg. 43-48, 51-56, 202-205) written by the hidden gaon Rav Tzvi Infeld Shlit"a.