If you want someplace super Unfancy, the Sea Shack has great seafood. And if I remember, a roll of paper towels on the table. It's very popular with the locals, so eat off-times or be prepared to wait.
We also like Dockside, Crazy Crab at Jarvis Creek, Old Oyster Factory, and Black Marlin (though I wasn't too impressed with our latest visit there). Those are all seafood.
We also like It's Greek to Me for....I bet you never guessed, Greek!
We like breakfast at Hilton Head Diner.
Since we live in Texas, and are 15 minutes from the Mexican border, we don't eat BBQ or Mexican at HHI. However, I've heard great things about One Hot Mamas and the owner and head cook was on the Food Network.