It seems like an obvious question, but it's NOT the canary. They're named after dogs, from the Latin root that gives us "canine."
== == The islands' name is likely derived from the Latin term Insula Canaria, meaning Island of the Dogs, a name applied originally only to the island of Gran Canaria. It is thought that the dense population of an endemic breed of large and fierce dogs, like the Presa Canario, was the characteristic that most struck the few ancient Romans who established contact with the islands by the sea.
The Canary Islands were named after the dogs ( Latin: Canis) that were present on the islands when the first Europeans from Spain arrived.
The islands were also the home of small greenish birds. When tamed and brought back to Europe they were called Canaria, and hence our word Canary.
So the islands were named for an animal, and the bird was named for the islands.
The Canary Islands are in the Atlantic Ocean and are named after the indigenous dogs of the island.