The correct answer would be A. Either Tom or Ken has his own band. Both Tom and Ken are males, and the sentence explicitly talks about them, which means that there can be no doubt that the pronoun refers to a man's band. If one of them were female, the C option would be considered grammatically incorrect; the B option would be acceptable, but the best option (from the point of view of grammar) would be to avoid a gendered form of the pronoun - therefore, the most acceptable option would be to say: Either Tom or Ken has a band. "Their" would be recommended only in the case of plural.
