Possible Clues:

• Mai-tai liqueur
• Blue liqueur, or the island where the fruit it's made from grows
• Ingredient in a Long Island iced tea
• Help a pig farmer, we hear, in Caribbean island
• Largest of the ABC islands
• Despicable sort I excluded from a farewell drink
• Thus a drink may make a so well, by the sound of it
• Resort island
• Spirit of the Caribbean
• Liqueur of bitter orange peel