I really wouldnt suggest renting mopeds...I was on a cruise a few years back playing crabs and the guy beside me was trying to win money to pay for the wrecked moped he just bought - he was pretty badly banged up. I'm not sure exactly what happened...i just know he had to pay for it. I've always been warned about renting those things.
I've grown up renting them every summer vacation at the beach and we have never had a problem. I know most of the cruise lines warn against them along with parasailing and jet skiing, but living in florida I don't know of anyone who has had a problem.