Would he enjoy a day of deep sea fishing? There are many boats that go out around the Bahia Mar area along the beach. There is also an all day pass on the Water Taxi ( $5, but heard it may increase) whre he can hop on ans hopoff all day , visiting spots along the water. Snorkling is another possibility, as well as scuba lessons.
Maybe a tour of the Everglades with an airboat ride? Usually a chance to see alligators and other fauna. There should be brochures around for guided day tours in most hotels- or try the yellow pages.
I keep having more brainstorms for your son- perhaps- ther is indeed a big movie theater right in Sawgrass Mall- I think it has about a dozen screens so he would have lots of choices- there is also a Gameworks Cafe right across from it- just full of all kinds of vidoe games etc. In addition, right near there is a branch of Ron Jon's Surf Shop- even though he is not a shopper he might enjoy browsing in what most guys his age find a very cool place. So, if he hung around Sawgrass while the ladies shopped, he could join you for lunch at Cheesecake Factory and pretend to admire your purchases.
Another thought on the other side of town, if he's "outdoorsy" is the complex south of the Airport that has the International Gamefish Association Museum, dedicated to fishing. I, as a female, found this not very exciting, but he might like it and right in the same complex is a giant Outdoor World Super Store which displays hunting, fishing, boating and camping gear of all sorts- lots of aquariums and displays and a big seafood restaurant- good hamburgers there as well. If you deposit him there , he might be able to fill up the better part of a day. Sorry things are so spread out around town- Lauderdale really is a city for cars!