First, the idea that the indigenous Taino and Ciboney peoples of the Caribbean were driven extinct is a myth, started by Bartolome de las Casas in order to help save the remaining population that still existed during his time. First, many of the first Spanish colonists came without wives and took indigenous women as brides. These wives instilled elements of their own culture into the upbringing of their children, which have persisted in the hybrid mainstream cultures of Puerto Rico, the Dominican Republic, and Cuba. DNA studies have confirmed strong presence of maternal Amerindian heritage in many parts of the Spanish-speaking Caribbean, especially in Puerto Rico.
Other indigenous people persisted by hiding out in remote parts of the islands, in the mountains and the forests, beyond awareness of colonial census takers. These communities took in fugitive African slaves and sympathetic Spaniards, developing "tri-racial isolates" similar to the ones that exist in the rural United States like the Red Bones, Brass Ankles, Carmel Indians, and Melungeons. One example is the "Yateras Indians" in Cuba, a rural society in the mountainous regions that has continually been identified as "indios" by mainstream neighbors.
In same cases, preservation of Taino culture was vital to the survival of the Spanish themselves. The first Spanish colonists needed to feed themselves, and the crops of temperate Spain weren't going to grow very well in the tropics, necessitating the adoption of Taino agriculture in the years before the arrival of African slaves. A look at Spanish Caribbean cuisine will show a fusion of indigenous ingredients and cooking techniques with African and European elements.
I suggest this fascinating book on the subject:
http://www.amazon.com/Indigenous-Resurgence-Contemporary-Carribean-Maximilian/dp/0820474886