Up to one liter of liquor or one case of beer per person. Extra is subject to import duty. You are also allowed up to one carton of cigs.
Do not bring any type of speargun, Hawaiian sling, or any parts for same. Importation of such is illegal.
Generally it is pointless to bring fresh fruit to the islands. Not sure if customs would seize it, but still it's not worth the bother.
If you are flying to the Brac on Cayman Airways then you may have a greater weight allowance than if you are traveling on Island Air. Regardless, I'd suggest that you only need a couple swimsuits, two or three pairs of shorts, and 3 t-shirts. Don't pack your entire closet.