The area around the Flamingo Visitor Center has a range of marked trails suitable for exploring by kayak. The Nine Mile Pond trail is particularly good, with plenty of alligators and wading birds. Sea kayaking is also possible with larger lakes accessible by boat.
In order to preserve the natural wilderness, there are no cabins inside Everglades National Park. Camping, however, is permitted and is a popular option during the dry season. Since you don't have to stick to designated campsites, why not take the chance to find the perfect spot to pitch your tent for the night?
If you're fond of pushing your physical boundaries, why not take up the National Park Service's triathlon challenge by picking up a Triathlon Passport from any ranger station? In order to complete your triathlon you must hike, cycle or kayak for 100 miles through the Everglades. The good news is that you don't have to do it all in one go.