Q. Why is the mitzvah called “Shiluach HaKen” (see beginning of Perek 12 of Chulin)? Do we send away the mother or the nest?
Perhaps, just as a person’s wife is called “his house” (see beginning of Maseches Yoma), and the reason has less to do with the fact that she is married than with the fact that everything that goes on there is a result of her sense of responsibility, her care, and her management; so too the “mother” of the birds is called the “nest.”

A. Adding to your interpretation, could be the words of Medresh Tanchuma (quoted in Rashi Devarim 22: 8) that teaches; “One that complies with the mitzva of Shiluach Haken Hashem says; If you have fulfilled the commandment of Shiluach Haken, you will eventually build a new house and fulfill the commandment of maako.”
Habonim Tikach Loch (p. 125) mentions that the understanding of the Rishonim (he quotes the Chinuch) is that Shiluach Haken actually means, Shiliach “Min” Haken, or sending “from” the nest. The word “Min” is missing, and this is often customary, as in Parshas Vaero; Betzeisi Es Hoir, that Rashi explains as “Min Hoir.”
Horav Shlomo Miller Shlit’a disagrees with the last interpretation since after all no “Min” or “Es” are written. The Rov’s opinion is that throughout Maseches Kinnim and many other Talmudic teachings, a “Ken” means a couple of birds, “Preida,” means just one. (See, Pada Es Avraham p. 86).

Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit’a