Why the play is called "The Tempest" and not "The Enchanted Island" or "The Magician"?

Playwrights choose their titles with care. A title must first intrigue the potential audience and second accurately represent the drama. While "The Enchanted Island" might intrigue its targeted prospective attendees, it would mislead them into thinking the play would be a delightful portrayal of a magical land full of fairy creatures. While the play is set on such an island, the plot is not as light and airy as such a title implies. If the play had been titled "The Magician," it would not be intriguing enough to make potential viewers want to attend. Such a title doesn't hint at whether the magician is good or evil or at what transpires.
"The Tempest" is an intriguing title that is meaningful in the context of the entire play. Hearing the word "tempest" makes one think of natural disasters and all the drama that can ensue. It also might refer to tempestuous relationships or emotions. A tempest suggests some sort of adventure. Londoners who knew the story of the Sea Venture, a British supply ship headed for Virginia colony that had recently shipwrecked in a violent windstorm, might be primed to attend a play about surviving such an ordeal.
The play's title is not a bait-and-switch. It begins with a real tempest at sea and a shipwreck, and that incident not only kicks off the action but also drives the plot forward as the separated parties deal with the presumed loss of loved ones and their plight on the strange island. There are plenty of tempestuous emotions as well, from Prospero's desire for revenge, to Antonio's murderous plot, to Ferdinand and Miranda's love affair. The play ends by resolving the negative emotions and restoring harmony as the characters sail on becalmed seas back to Naples.
The title "The Tempest" holds up as an intriguing and apropos description of the play's genre and plot.