What are the ages of your party? There is Aerial Hilton Head, Zip Line Hilton Head, Lawton Stables, shopping the outlets, biking, enjoying the beach/ocean, swimming in the pools, kayaking, paddle boarding, sunset cruises, dolphin cruises, fishing, etc, etc . Savannah is ~45 minutes away and a very easy day trip. Charleston is further but you could spend a day there as well.
The thing we like to do most at HHI is relax.