Buachaillí is the actual word. In irish, in certain circumstances we use séimhiús (h) eg after mo, do or an. In different circumstances we use úrus (changes depending on the word m for b, n for g etc). Úrus are used after ár, bhúr and a (meaning all of them).
Eg Tá buachaillí ag imirt peile. Tá mo bhuachaillí ag imirt peile. Tá ár mbuachaillí ag imirt peile.
As far as I know, the two other options stand for some modificacions (lenition) produced by certain words before a noun. Since there aren't any here (fortunately, because it hasn't been explained in any lesson yet :P) the correct one is the option with no lenition modification :)