Yes, as long as the name is not offensive, individuals should be allowed to name islands. Owners should definitely have the right to do so, while regular citizens should be able to name unnamed islands by the same principle as explorers in the past earned the naming right upon discovery. If amateur astronomers get to name meteors freely, why wouldn't an unclaimed island be the same?
Individuals can name an island whatever they want to, but that doesn't mean that's the name that shows up on maps. Whoever has jurisdiction over the island should get to name it. For Busta Rhymes Island, Massachusetts residents should figure out who had jurisdiction over it first before it dishes out any naming rights.
While the Shrewsbury, MA island may always be referred to as Busta Rhymes Island to some locals, it is best that official names go through a review process to ensure the name is not discriminatory or insulting to some people. An island's name will be attached to it indefinitely and must be taken seriously.
In order to name an island, a person should have some sort of stake in it. Why should one random person just decide he gets to name an island? Someone has to be in charge of recording that sort of thing. I think if there was a public contest, that would be different.