Hello, Keirsten!
You're going to love how quaint, quiet and homey Disney's Hilton Head Island Resort feels from the moment you step onto the property!
When we visited with our then 4 and 7-year olds, we really enjoyed The Salty Dog Cafe, which is located at South Beach Marina Village, in Sea Pines, a short drive from the resort. We had a great table outside by the water, which afforded us some deliciously fresh seafood, and we didn't have to worry if the kids spilled some of it on the floor. Paige, in Reservations for Disney's HHI, put me in touch with the resort's front desk for more options.
Christa, at HHI's Front Desk, also highly recommends Skull Creek Boathouse, The Crazy Crab and Giuseppe's Pizza and Pasta for families who are looking for great food at kid-friendly places. All of these have delicious menu options, a kids' menu and wonderful atmosphere. BTW, Giuseppe's can also deliver to the resort!
Christa also told me that there are 23 golf courses on the island, and that Palmetto Dunes, which is right across the street, is very kid-friendly (they even have Junior Rates!)
For tennis, Palmetto Dunes also has 23 clay courts and 2 hard courts, with a fee that's about $20 (at the time of this writing.) However, Christa also mentioned that Chaplin Park, about 2 miles up the road, has complimentary courts.
Sounds like you are in for one terrific stay at Disney's Hilton Head Island! Please send any additional questions you may have as you plan this great trip! I would love to help!