Amîrah is indeed an Arabic name meaning "princess, leader" according to
Digest of Muslim Names by Fatimah
Suzanne al Ja'fari, and the appendix dealing with Arabic names in Hanks & Hodges's
A Dictionary of First Names lists Amîra with the same meaning.
I don't think this would be pronounced as "Am-rah", however. It's more likely "ah-meer-ah".