Ain means eye or spring/fountain. Aini means "my eye", or "my spring". I haven't seen this used as a name, but there is no reason why you shouldn't. If you put the stress on the end, i.e. Ayniyy, it means "similar", "of the same kind".
Aini is not used in the Quran, but its root is AIN-Y-N which is used, both to mean eye and to mean spring/fountain. For this reason Aini can be considered an indirect Quranic name. And as said, since the meaning is acceptable Muslims can use Aini (and Ain as well) to name their babies.