The Caribbean is named for a people, and the name did not start as theirs.
Etymonline traces "Carib" to Spanish Caribe, which came from an Arawakan name used in the West Indies. Wikipedia says Christopher Columbus was the first to record the word. It is called an exonym, meaning a name outsiders gave. The meaning is given as "brave ones" or "strong men," and Wikipedia mentions "brave warrior" as one hypothesis.
English took "Carib" into the language in the 1550s. The adjective "Caribbean" is built from it with the suffix -ean, and Etymonline dates it to the 1750s. Etymonline lists two senses from the same decade: "of or pertaining to the Caribs," and "of the sea between the West Indies and the South American mainland." The region as a whole took the name from the sea.
The people themselves used other names. Wikipedia gives the language's own terms as Karifuna for one person and Kalinago for the group. In 2015 Dominica officially changed its usage from "Carib" to "Kalinago." Wikipedia also cautions that not every Indigenous group in the Lesser Antilles should be called Kalinago.
There is a darker thread. Etymonline says "cannibal" comes from Spanish canibal, based on Caniba, the name Columbus recorded for the Caribs. Wikipedia reports that stories of flesh-eating Kalinago reached Columbus secondhand, and that there is no hard evidence the Kalinago ate human flesh. The historian Hilary Beckles has argued the stories served as an ideological campaign to justify colonial expeditions. So one Indigenous name gave English both a region and a word for a horror that may largely have been invented.
The name outlasted the era that produced it. Spanish explorers used "Carib" for groups they saw as hostile and "Arawak" for those they saw as friendly, a split that said more about the Europeans than about the islanders.
Languages of the region left other words in English. Etymonline traces caiman, curare, manatee, plantain and potato to Carib languages, and mangrove perhaps to one.