the island is called Elliðaey.

as best i can find for an actual reputable source, they only thought about giving her use of the island, but i'm not sure if it actually happened.

Earlier this week David Oddsson, Prime Minister, told parliament: "Bjork has done more for the popularity of Iceland than most other Icelanders. My view is that she may be given the use of this island [Ellidaey] as a royalty payment, as recognition from the state."

But, amid an uproar in the country, the government has had to reconsider its generosity. Yesterday Mr Oddsson's office denied Ellidaey had been sold or leased to the singer.

source

according to wikipedia, it is uninhabited