I have a student and a friend, both named Cáit - pronunciations here (Cork) vary from Cawt (caught, taught, thought) to Coit, both with a slightly softened T. My friend Caitlín flattens the first vowel out slightly, but I think that's just a concession to us blow-ins who are deaf to the tonal subtleties (she's a fluent Irish speaker and singer). As with so many other words, this pronunciation probably varies from region to region. Packie (an Ulsterman) used to say that Munster Irish was nearly a foreign language to him.

-lín would be Lean, and the name is Irish for Kathleen.