I would suggest Key West. It’s very family-oriented these days.
Things that kids might like:
Shipwreck Museum
Treasure Hunting Museum
Lighthouse
Aquarium
Conch Train (like riding Mr. Roger’s Trolley around Key West)
Mallory Square (on the waterfront, has lots of jugglers, musicans, artists, clowns, etc etc )
Glass bottom boat rides (see the coral reefs)
Bike Rentals (rent bikes with child seats)
Ripley’s Believe It or Not Museum (might be scary for some kids, was scary for me)
Beaches – Smathers Beach is sandy and full of volleyball and practically naked people. Ft. Zachary Taylor Beach is a bit rocky, but with sandals it is good for kids because the water is beautiful and it is not very crowded.
Restaurants fun for kids:
Turtle Kraawls – on the waterfront, depending on the time of year they have various marine life in tanks. Mexican, burgers, salads, etc.
Blue Heaven – cafe type restaurant … this is great for breakfast. Dining is out-of-doors in a barnyard type area with baby chicks, geese, and kitties running around underfoot. Personally this gives me the heebie-jeebies – I might be eating EGGS for goodness sake, but it was a hit with the kids.
Most of the larger hotels have some sort of day-care or supervised children’s activity in the event you and Mrs. Rundogrun want to knock back some margaritas at Sloppy Joe’s one afternoon.
I love Key West because it is warm and tropical, yet it is easy to get around. At the risk of sounding like a complete Philistine, there is a Blockbuster, K-Mart, Walden Books, Super Cinemas etc etc for all the things you will no doubt have forgotten. Also conveniently has things like Pizza Hut in case your kids want something generic to eat.
You do not absolutely need a car in Key West, but with kids I would recommend one. Especially if you are staying in a condo unit or other apartment-style accomodation, you will need a car to get to the market.