Muscadine grapes are a specialty in Florida. Not sure why no one here clarified that there are in fact warm-weather grapes, and specifically muscadine grapes which are a different species that deal well with our conditions.
I second also the recommendations of various vining vegetables. The only downside with a few of those is if you have iguana problems... for instance Malabar spinach grows vigorously here, and is very attractive, but is eaten right to the ground by the damn iguanas. Garlic vine is also a good one, with pretty purple flowers.
Scarlet runner bean is a vigorous grower if you want red flowers.
Bougainvillea is also edible in salads (the colored flower part), so I find that to be a winner because it's easy to grow here, drought-tolerant, covers fence, comes in many colors, and is very attractive. Not everyone likes the thorns, but some people prefer them as property protection.
Mysore raspberry is another option if you want berries specifically but also like thorns - those are some serious thorns!
If you don't mind the look, you can also use the fence to prop up dragonfruit.
A lot of your decision will really depend on your local conditions, and whether the fence section is full sun, full shade, or partial sun.