Are you sure it is a *name* from India? I haven't heard it there.
The word bAi is used in India often as the last part of a name, or as an honorific title. It also has the connotation of a dancing girl in some other circles. The etymology is from Turkish bAjI which again is used as an honorific or meaning elder sister, but since I know no Turkic, I can't tell you more.
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