I don't normally like religious names, especially names that are as religious as Adah, but I find the name Adah extremely beautiful and I would imagine a female named Adah to be lovely.
My daughter's name is Adaya ( ah-day-ah). It's a variation of Adah in our baby name book. It means "a beautiful addition to the family". We love it and everyone who meets her loves it too!
Adi is more common nowadays.