Not weird but not widely served in the tourist areas:

Run Dung is the best. My dad makes it all the time but my favorite kind which he can only make while in Jamaica is Breadfruit Run Dung -- it's made with young breadfruit and salt fish.

Chocolate Tea --- basically hot chocolate but it is grated raw chocolate

Cornmeal Pudding -- with the slushie on top

I have yet to try blue drawers but it is on my list of have to try.