Hello from Spain.
This name is quite uncomon in Spain but there are a lot in Cuba. Think on the musician Beny Moré. His ancestor was a slave of a certain Moré. when manumisio arived the black got the name of his former master. And like him many more blacks, maybe hundreds. Thus, there are a lot of Moré in Cuba. Just check the telephone listing. So, if you are black or mulatto, your origin could be there.
If you are white, then consider that this Moré family (the slave master) was quite important in Cuba. Around 1850 they even got the nobility title, there is a Count of Casa Moré, quite powerful and prospanish during independence wars. It was a popular name, and many people simply adopted it. There was not social security numbers and cards in that time. When arriving to Cuba, if you wanted to be called anithing christian, nobody could tell you otherwise. Many people did just that, the same as when somebody arrived to Ellis Island. But there it was also a linguistic problem. Maybe your languaje was not understood there and they simply called you Peter.
If it is not your case (that could be in 15%) then you shuld chek in Cuba. I can tell you that i'm not related with you. Sole posibility is a illegitimated relation with a later legalisation without knowledge of my family. Very unlikely. If you know the town of your encastors in Cuba, check the local church registry. Everithing should be there.
can provide you more info in mercadosemergentes@hotmail.com