Sea Pines Resort on Hilton Head Island, SC is a great young child destination if you are looking for the "lazy days let them play" type of place that is still great fun for adults. It can seem pricey when you first look at it, but a closer look might change your mind. Accommodations range from hotel room like small units to large private houses. Some are for rent from the private owners and others through the resort. Peak tourist season is expensive, but the weather at Sea Pines makes off peak times very enjoyable.
You can find a house to rent privately directly from the owner, some for as low as $900 for a week off-peak - well below hotel costs. These can sleep 8-10 or more and full kitchens mean you save a lot of money by eating in most of the week even with admittedly higher grocery store prices. Most houses come with internet, cable, dvd players, etc. They have nice kitchens and decent linen service. Some owners provide game rooms, beach toys and more. Some homes have private pools or hot tubs.
Bike rentals are around $20 for a week (there is a place not far outside the resort that has coupons in all the tourist handout mags there that makes them even cheaper) and child seats or trailers are only a few dollars extra. There are over 30 miles of bike paths within the resort, plus the beaches, and they are some of the best groomed trails I've ever seen. The bikes are large frame with big cushy seats and the trails are wide, smooth and mostly flat so you don't have to be a habitual cyclist to enjoy it. You can rent bikes and basically leave your cars parked at the house. The trails lead to all the places you'd want to go; the beaches, a little nature center and larger nature preserve that offers programs some of the year, Harbortown shopping and restaurant area and marina, Harbortown playground - large with several different structures plus a huge tree house, public pools, fishing ponds - some designated for children only, golf courses, historical sites, etc.
Wake up in the morning, eat breakfast, bike to one of the above mentioned places, let the kids romp and play- build sandcastles and fly kites, whatever, then have a picnic lunch, bike back for naps, hit another 'lazy' destination in the afternoon, make dinner at the house, put the kids down for a slumber party. Adults have all the typical resort amenities and excursions available limited only by your pocketbook. Rainy day? Just outside the resort is a hands-on children's "museum" called The Sandbox. Admission is $5 for each person over 2 but I believe they publish coupons and have special events for free or reduced cost. The kids can spend all day playing in the space shuttle, a pirate ship, the arts and crafts room, constructions site, etc. Also, Hilton Head and the surrounding area has lots of arts, nature and sports available - from museums to sea kayaking (you can do this even with toddlers) to nature walks to horseback riding to historical tours. Again, peak season is expensive, but off season the weather is balmy, the prices are reasonable and the trades have lots of coupons.
The only drawback we have found is the price in getting there. None of the low fare airlines, like Southwest, fly into Savannah or anywhere else close by, so you have to troll the fare watcher boards to get good airline deals.
Hope this helps!