Well, I don't know about what everyone else is posting- where I come from, most people would pronounce Regan as RAY-gun, not REE-un. It doesn't mean "regal and royal" OR "little king", but I see everyone else has already told you that. It's always more fun for sites to give meanings for every name, even though a lot of them are of unknown meaning. Makes the expectant parents think the name is more solid, I guess.
I myself prefer the RAY-gun pronunciation, BTW. It has nothing to do with Ronald Reagan, I just like the sound of it.