If Jewish parents adopt a non Jewish boy and he converts. What is his Jewish name? Example Moshe ben ?.
Essentially his name is ben Avrohom Aveinu. However R’ Moshe Feinstein in his teshuva where he discusses the various aspects of making the bris and calling him a name, he says that the child can be called by the name of the adoptive father. He says that there is no concern that he might marry someone from his adoptive family, because the parent have to inform him before his bar miitzva that his parent are gentiles, therefore we are not afraid of that.
Dinei Hagiur Vhager pg. 51, Igros Moshe Y:D 1-141, Mishne Halachos 3-26.