For the sheer number of activities you can do in the Cancun region, it's a shame that the ship only spends a day in Cozumel. Here are some of my suggestions.
Culture/Sightseeing:
- Chichen Itza is by far the best of them all - one of the 7 new wonders, as you may know. It's a full day experience, and you will have to do it with Disney to not risk missing the ship. The site is absolutely worth it - but if you do plan to visit Cancun on your own at another time, you could save it till then. There are a couple of other amazing stops on this trip - the town of Valladolid and the Suytun cenote - that the ship excursion won't have time for.
- Tulum is tiny in comparison - doable in one hour - but it's much more accessible. The Tulum beach is also known for its powdery white sand. There are a few other minor ruins as well - Coba, Uxmal, Ek Balam, and El Rey - if you want to split your day with something else.
Water Activities:
- Yucatan peninsula has an amazing array of underground rivers and swimming holes called cenotes. The best way to experience them is to go for a swim at the Xcaret Park. I will let you google the details on your own, but Disney does run an Xcaret excursion. The park is an adventure of its own.
- Xcaret has other sister parks too - check out Xel Ha, Xplor, and Xavage.
- Our own favorite water activity in the Cancun area is a walk on the sea floor (!!!). It's called SeaTrek, and you can experience it right there in Cozumel. (Disney also offers an excursion.) Or, you can do it at the Xcaret Park. The activity doesn't take long, and you can easily combine it with the river swim.
- If you are after a waterpark, check out Jungala Aqua Experience near Vidanta resorts. And if you want to combine a few water activities in one day, look up Aquaworld Cancun.
- If you are after snorkeling, do it in Cozumel. Cozumel has some of the best reef in all of the Caribbeans. Disney offers a few excursions that include the activity. If you can swim underwater, the Snuba activity (= a cross between snorkeling and scuba) is a lot of fun.
Resort/Beach Day:
- Book a (day) room at any of the resorts in the Hotel Zone. JW Marriott IMO has the best pool complex, while Ritz the best service. As a bonus, there is a lot of shopping and dining nearby at the Isla mall and surrounding markets. You can also do parasailing, jet ski, speedboat, and a bunch of other activities right at their beaches.
- Get a day pass at the Moon Palace. The pool complex is the largest in the area, and they have a waterpark on site. The resort is outside the Hotel Zone though, so the beach is very modest.
- Disney offers plenty of beach-day excursions in and around Cozumel. Look them up.
Other:
- A very unique way to spend you day is to visit Isla Mujeres - just north of Cancun. Buzzing with laid-back vibes, this small island has an archaeological site at its southern tip, and a world-top-20 beach (Playa Norte) at the northern one. What's really cool though is that the island hardly has any cars, and pretty much everyone gets around on golf carts - which you can rent too!
- If you don't want to get wet but still want to explore the reef around Cozumel, do the Atlantis submarine expedition. It doesn't take long, and you can combine it with something else.