This is a really cool (and educational) option right next to the resort, especially for kids. From the website: Georgia’s only sea turtle education and rehabilitation facility. The Center offers the public a chance to learn about sea turtles and see rehabilitation in action with a host of interactive exhibits and experiences.
Any of the tours on Jekyll Island would be great daytime activities, but we espcially recommend the Landmark Trolley Tour for a quick history of the estate. It's a fun way to see the historic district with guided narration on the Gilded Age moguls who created the Club, and you're also given the chance to look around in one of the historic cottages.