An uninhabited island off of the coast of Kingston and Port Royal, Lime Cay is a hidden wonder of gorgeous white sand, sunshine, and crystal-clear waters. A 15 minute boat-ride from the mainland, this tiny island has no tourist infrastructure and yet still remains a highly popular retreat for the local weekend crowd, making for some great impromptu parties and beach recreation, especially on Sundays. Rent a boat in Morgan's Harbour or, better yet, arrange with a local fisherman for drop off and pick up.
If you're looking to stage your own private paradise, then grab some snorkeling gear and head out during the week, when it could quite possibly just be you kicking through crystal clear waters in search of varied marine life (kingfish, coral, barracuda, etc.). It'll be a great change of pace to the over-burgeoned tourism of many of Jamaica's more popular attractions.