For most of the colonial period, the Caribbean was was considered the most valuable real estate in the Americas by most colonial powers. Why? Because the cash crops being grown there, particularly sugar, were so incredibly valuable. Now, this offers both problems and AH potential.
The problems are those inherent to an economy based entirely around agriculture, particularly a single cash crop. Namely, the region is dependent on said crop, and in the event of some change, is not really capable of responding. For the caribbean, as other markets develop, new crops (and substitutes for their products) emerge, and as economic systems change (industrialization, end of slavery, etc...), the system isnt terribly well prepared to cope. Simply put, once the sugar trade isnt producing the same profits, there isnt a whole lot left.
Now, there is an upside to this. remember, the caribbeanwas, in european eyes, prime real estate. So, in theory, were a european nation to invest in the islands, and see them as something beyond a source of cash crops, then the region could diversify, and thus weather changes better. The problem is, what do they add? there isnt a whole lot of land or resources, so many industries or large-scale immigration are out. But having the european nations focus on developing the islands, not just the crops, might be the best bet.
One other thing. If the POD is a relativly late one, and involves at least partial de-colonization, the region has to be kept safe from foreign meddling, particularly that of the united states. it isnt good for independence, stability, democracy, or a high standard of living.