The last time we were in Grand Cayman, we booked a shore excursion thru the cruise line which took us to the Beach Club on 7 Mile Beach, the price of the excursion entitled us to a lounge chair (packed so close to each other you didn't want to disturb everyone to get to the water) and a drink from the bar (tasted like red Kool-Aid...no alcohol), they had everyone jammed into one small area. We packed up our stuff and left within 10 minutes of our arrival and complained to our shuttle driver. He suggested next time we visit Grand Cayman that we hop on a bus at the pier terminal and ask the bus driver to drop us off at a public entry point to 7 Mile Beach. Sure would have saved us a lot of money and aggravation.
In Cozumel, we have booked excursions to Playa Mia and Chankanaab Park which would be quite costly for a large family. I'm sure other posters can help you out with cost saving ideas. A large shopping area is right at the end of the pier so you don't have to travel far to pick up a few souvenirs and I understand there is a Margaritaville there where you can just order soft drinks and appetizers and kill a lot of time in their pool.
The last time we were in Ocho Rios, we just walked to the shopping area and, again, availed ourselves of the amenities at Margaritaville. My grandson loved swimming in the pool and didn't want to leave. The big draw in Ocho Rios is Dunns River Falls but quite costly for a large group.
In Cozumel Paradise Beach is a good choice. Cab ride is about $15 each way. No entry fee to beach. For $10 each the kids can play with the water toys all day while Mom & Pop lay in the shade. Umbrellas, chairs, clean restrooms, shower, and changing area all available for free. Food and bar service available.
In Grand Cayman there are taxis (vans) that will take you to the beach. Seems like cost was $4 or $5 each. One of the tours to Stingray City is another good option. They can be purchased local or thru the ship.
In Ocho Rios Dunns River Falls is great but I would be concerned about taking a 5yr old on that trip. For a first time in Jamacia I would suggest sticking with the ships tours. It is not as tourist friendly as your other two stops.
In Grand Cayman, the best rate I found for a great trip is on The Hannibal Stingray City Catamaran its only $35 for adults and $27 for kids! Transportation is included and you can bring your own snacks on board if you want.
In Cozumel our favorite place is Nachi Cocom Beach Club its all inclusive of drinks and food, with pool and incredible beach!!! It only $12 for kids for everything, for adults its $49 but so worth it, the food it great and the drinks are top shelf, can't even remember how many margaritas we drank;-)