I just wanted to know if it is permissable to keep the name amira (meaning) leader princess for a baby girl? or the name amaya (meaning) night rain for a baby girl?
Praise be to Allah.
Parents should choose good names for their children that do not go against the teachings of Allah and that are not regarded as odd in their environment or according to their customs. They should also take care to ensure that the name will be suitable for all stages of the child's life and will not be a cause of annoyance for them. Imam al-Maawardi ash-Shaafai (may Allah have mercy on him) mentioned a number of matters that it is recommended to pay attention to when choosing names, among which is the following: (The name) should have a good meaning, be appropriate for the one who is given this name and be one of the names that are customarily used among the people of his class, religion and status.
End quote from Naseehat al-Mulook, p. 167
With regard to the name Amirah, we do not see anything wrong with calling a girl by that name, because it is not contrary to Islam.
With regard to the name Amayah, this is not an Arabic name; hence no Arab should give his daughter this name, because it is somewhat odd and is not commonly used among the Arabs as a name. But if the one who wants to give this name (to his daughter) is not an Arab and the people around him do not find it odd, then it is permissible for him to give his daughter this name, so long as the meaning is acceptable and there is nothing in it that is contrary to Islam.
See the answer to question no. 1692 for a discussion on names for children that are haraam or makrooh.
See the answer to question no. 101401 for a list of good names for females, along with their meanings.
And Allah knows best.