A typical western Caribbean itinerary includes both Cozumel and Grand Cayman. Both have good programs, either through the ship or on your own, for novice and advanced divers. I'd stay away from diving in Jamaica.
A typical eastern Caribbean itinerary includes St Thomas and/or St John, and some include San Juan, PR. I recommend Low Key Watersports in Cruz Bay, St. John. Great dives for novices.
Southern itineraries out of San Juan all include lots of islands with good diving, but I haven't dived any of those islands except Aruba. Red Sail Sports has a good program there.
Roatan and Belize both have outstanding diving, but I don't know that I'd recommend it for a novice.