I live 30 minutes from HHI and I can tell you there are no amusement parks anywhere near us. There is a children's museum about 3 hrs away, and aquarium about 2 hrs away. Carowinds is in Charlotte, NC but that is 4-5 hrs away. Mostly, we have beaches, golf, and historical tours. Hope this helps.
Carrowinds is alot of fun and has a wonderful water park, which is included in the ticket price, however, it is about 4 hrs from HHI. I don't think I would try to make it a day trip from HHI.