Call back HP and make a follow up on the call you made sometimes things get lost in translation, usually from the item leaves the island they don't attend to it but since from your description above they are aware that you are in Jamaica and suggested/offered for someone to come and pick it up to repair then you should follow that up. You may also want to check Royal Computers since they seem to have a contract with HP now so they may be your best place to check next if all of this falls through.
Intel i5 2300, Gigabyte Z68XP-UD3, 12GB DDR3 @1600 MHZ, XFX 6870 2gb (Stock). All Stock
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