I find it funny that people find this important. It is more symbolic than anything. After all, the Queen actually holds no power whatsoever, even in England.
Jamaica is an independent country with or without the Queen being the head of state. Making this change is not important and is purely a political rouse to gain votes.
Getting rid of the Queen by itself will have no effect on the actual running of the country. Now, on the other hand, the creation of the position of President is what would actually make a difference. I think what the Prime Minister is really after is being able to be elected as President (typically a more powerful position than a Prime Minister) and getting rid of the Queen is a convenient way of gaining sympathy for his real cause, gaining himself a position that has more power.
The position of President has its good points and its bad points. He is more answerable to the people since he is elected by the people, but he also holds more power than a Prime Minister would and can make decisions regardless of the what the people or Parliament thinks. He might not get re-elected, but he has the power to do what he wants for his term (to a limit of course).
A Prime Minister, on the other hand, is not answerable directly to the people, he is answerable to the Parliament which is elected by the people. Also, the Parliament can remove the Prime Minister if they don't like his actions. This limits his power since he can't upset the Parliament too much or he will face the ax.
I think the real question should be whether or not Jamaica should have a President or a Prime Minister. Whether or not the Queen is head of state is actually irrelevant to how the country is run.