I think it's a beautiful name, but don't like the possible connotations of the meaning (taken from an indian website):



Arabic amr, from which comes the word amIr and its contracted form mIr, which was a fairly popular name for Indian Muslim males (Mir Qasam, etc.). The feminine form mIrA would be a purely local creation.

amIr means "noble, prince, lord, person of rank or distinction," etc. (Cf. Platts Classical Urdu dictionary).

The derivation of bAI as coming from bhavatI, the Sanskrit polite form of address to a woman. Unfortunately nowadays, the term is somewhat pejorative ("dancing girl, harlot"), especially as bAI-jI.