We always eat at Pancho's Backyard
in Cozumel and it is heavenly! It's in the courtyard of Los Cinco Soles (one of the recommended shops that will be on your Cozumel map). The atmosphere, food, and drinks are just wonderful. One of the main reasons we are going on a Western Caribbean next is JUST so we can go back to this fabulous restaurant again.
They claim to have their own in-house purification plant so you don't have to worry about water or ice. We've been fine every time we've eaten there.