I'm a part-time HH resident. We absolutely love it there. December can be nippy, but still nice enough to go out and do things. Bike riding and walking on the beach were already suggested, my favorites. Golf and tennis as well. How about a kayak trip for the kids? Outside Hilton Head has activities that leave from Shelter Cove, next to Disney. There are also trail rides at Lawton Stables, located in Sea Pines Plantation. When you get to HH, grab an activity guide (free, published monthly) to see what special events are going on. You also can't go wrong with a sunset dolphin cruise.
For "cheap" seafood, there is always Captain's Seafood. It is part fish market, part restaurant, very reasonable prices, very fresh. OTOH, they keep jazzing the place up, so the prices have gone up to match, but it's still a good deal. Another route to go: early bird specials. Don't laugh--there are some wonderful meals to be had at fantastic prices---pay about $15. for a 3 course meal at a place that runs $25. per entree an hour later. Check out the restaurants guide (also free) when you get there, for menus and prices. There's also a website called "10Best"---it has never let us down.
HTH and have fun!!!