DH and I stayed at HHI over the Thanksgiving 2006 (Kind of).
We arrived on Friday after Thanksgiving and stayed until Monday morning.
I believe you could order a Thanksgiving meal from Disney or from area grocery stores like Piggly Wiggly. Others can tell you since we were in Atlanta with Darling MIL!
Activities while there included: Goofy Bingo, Campfire with S'mores, Storytelling with B'lou Crabbe, ornament painting, kids activities, dolphin tours, and kayaking. Big Murgie's den is a great place to go and relax by the fire, surf the web (if you have a laptop) and play lots of billiards. (I may have forgotten something.)
The weather was a bit cool, but nice.
Savannah, the Hilton Head resort, and the beach were all very romantic! It was a very nice adult vacation. However, the biking and activities would make it nice for kids too!
It was probably way too cold to swim!
Overall, we liked the vacation very well. You might want to do a DIS search for hilton head and thanksgiving. There are other threads about it here.
We went in 2006, and we already have ressies for 2007. Yes, you can order a turkey dinner from the resort and it really smells nice on Thanksgiving morning as they deep fry turkeys outside. You can also order a turkey dinner from Piggly-wiggly, which is what we did. Also, there's a big beach football game (DH wanted to watch the Dallas Cowboys and the time conflicted, so we missed this). On Thanksgiving morning, there's a craft time for the members of the two beach teams to make their tee-shirts. We had a wonderful time! But beware-make your ressies early-it's a pretty busy time at the resort. The weather can be very varied as witnessed by these two pictures we took that week:
That's steam rising from the pool because the air is so cold.