When I asked a question recently about a drasha for a specific din, I had some trouble choosing the tag for "drasha." In the end, I chose exegesis, but then realized that the only other question with that tag was Baby Seal's Pardes question, which is more of a question about "drasha theory" then about specific drashos. A chat discussion with DoubleAA led me to use the midrash tag.
But....I still don't think that this is the right tag, because a cursory look at the question that fall under that tag reveals that some are questions about specific drashos [system of exegesis] (1, 2, 3), while others are questions about the Midrash [books that explain Torah, such as Midrash Tanchuma, Toras Kohanim et al] (1, 2, 3).
Therefore, I think that the midrash tag should be split in two: with questions about aggadta and the books Midrash staying with midrash, and a new tag ("drashos"? "drashos-exegesis"?) for questions about specific, halachic, drashos.
The current tag description for the midrash tag is "Exegetical interpretation, often of verses in Tanach. The term also refers to books based on these interpretations, e.g., Midrash Rabba, Midrash Tanchuma, et al." .....I think that we should leave the second half of the description ("books based on these interpretations, e.g., Midrash Rabba, Midrash Tanchuma, et al.") with the the midrash tag, and the first half ("Exegetical interpretation, often of verses in Tanach.") with the new tag.