From Middle English ametist, from Old French ametiste (French améthyste), from Ancient Greek ἀμέθυστος (améthustos, “not drunk”), from ἀ- (a-, “not”) + μεθύω (methúō, “I am drunk”), from μέθυ (méthu, “wine”).
The name was based on the ancient belief that amethyst gemstones could prevent intoxication. People in ancient Greece and Rome would wear amethyst or drink from cups made of amethyst to ward off the effects of alcohol.