The -bAi bit is easy: in the languages of western India (Hindi/Marathi etc.), it is a feminine honorific. As a relation term it means mother in some languages. It also has the derived meanings of a lady of art, a dancer, or prostitute in some languages.
The Meera (as it is sometimes spelt: both vowels are long) bit is more complicated. I do not understand where the meaning `prosperous' given on this site comes from: I am not familiar with this meaning in Sanskrit. The closest to mIrA I can find in Sanskrit is mirA, a limit, or mIra, an ocean, and I think both of these are unrelated. I suspect the name may actually have connection with Farsi mIr meaning main. (My Sanskrit dictionary claims mIramIrA is a woman's name, but it does not provide that with a derivation of a meaning.)