We went to Phuket, Thailand for our honeymoon. The guide told us that this name is actually a corruption of the Malay word, "Bukit" which means "hill", because the island is so hilly. And as to why a Thai island has a Malay name? It was part of the Pattani Malay kingdom that was conquered by Thailand during the colonial era.
After so many generations of Thai rule, there are still many Muslims there. However, very few can still speak Malay. Further south nearer to the Malaysian border in the province of Narathiwat, however, I am told that many Thai Muslims can still speak Malay.