Assuming your cell is unlocked, you can buy an inexpensive sim card from the digicell noted above (around $20 - $25). Then it's simply purchasing Jamaican minutes which are inexpensive. The service and phone number will be from Digicell, not your home provider. Therefore no roaming or data charges from your home provider.

Many providers lock your service because your in a contract in exchange for a break in the phone price. That's understandable. Most (all) will be unwilling to unlock your phone. However, there are "grey market" websites that will, for a fee of around $15, provide a "key" and instructions on how to unlock your phone.

Your other option is to rent or buy a Jamaican phone. It will already have a Jamaican sim card so you simply purchase the minutes you require. Some airport transfers offer a phone when using their service. Give them a $50 deposit and but your minutes and it's yours to use for the duration of your trip. I belive, Alfred's and Rocky's both offer this as part of their service to you.