We've rented a scooter in Cozumel, once. We bought their "insurance" to insure we were not railroaded. Dave posted a link to an article about a preacher that was jailed in Cozumel for a "fake" accident when they rented a jeep. If you're going to rent a vehicle, make sure you purchase any and all insurance policies they own. This is a foreign country and our laws do not exist. Fairness/trials/legality are only suggestions and followed only by those that want to.
That being said, there is plenty to do in Cozumel on your own. Chankanaab, Paradise, the Money Bar, Nachi CoCom, etc., can be accessed by taxi. Some charge entrance fees, some charge cover charges, but all are easy to research on the internet.
Just be sure you are back to the ship in plenty of time. If you're not on a ship's excursion, they will NOT wait and it is your responsibility to catch up to the ship. If Cozumel is your last port of call, you'll miss out the last few days on board because your next port of call will be your embarkation port. I'm not trying to scare you, but just stating facts. I've seen far too many standing on shore while the ship is underway.
Just be safe and make sure you've done your research on any port before trying to do it on your own. Also, make sure you speak the native language. Although many people in Cozumel speak English, many do not.