Combine rum, lime juice, onion, ginger, garlic, allspice, thyme, cayenne, salt, and black pepper in a blender and pulse until smooth.
Lay chicken thighs in a large casserole pan and pour marinade over the top. Turn chicken to evenly coat. Cover with plastic wrap and let rest in the refrigerator for 8 hours or overnight.
Prepare the grill by cleaning it and using a small amount of oil to grease grate to prevent sticking. Turn the flames on to medium.
Grill chicken thighs over medium heat, skin-side down for about 10 minutes, then flip and continue grilling for another 8-10 minutes. Smaller or boneless thighs will take less time to cook.
Notes
You can swap the ground cayenne or ancho chile for 1 diced habanero or Scotch bonnet pepper. Be sure to wash your hands thoroughly after handling! If you like, reserve the marinade after removing the chicken, bring to a boil in a saucepan, and use to baste chicken.