West Indian island, from Taino (Arawakan) xaymaca, said to mean "rich in springs." Columbus when he found it in 1494 named it Santiago, but this did not stick. It belonged to Spain from 1509-1655, and after to Great Britain. Related: Jamaican.
The Jamaica in New York probably is a Delaware (Algonquian) word meaning "beaver pond" altered by influence of the island name.