OK, as a name nerd, I have to freak out here and be annoying. First of all, the umlaut on the "e" is weird. It's not used in any languages I know of except Albanian. When used in names like Zoë, it's to make it so the E is pronounced when normally it would be silent.
Regan is an anglicization of the Irish surname Ó Riagáin, which does stem from the word "rí" meaning "king" and a diminutive ending.
As a first name, Shakespeare was the first to use it; who knows what he based it on.
Thus said, I like the sound (REE-gan, not so fond of RAY-gan). I probably wouldn't use it, but it does have a nice cadence, and I did like "The Exorcist."