The Falkland Islands are in a seriously remote area of the South Atlantic. Closest land is Argentina, but it's still a very long way.
The Falklands are cold and windswept, but big enough to have a population, a couple of small towns, and a few thousand sheep. Even more remote are places like South Georgia, which are uninhabited, apart from penguins.
These islands are all owned by the British. You may also remember the Falklands War in 1981, when Argentina invaded them to try to claim sovereignty. Lots of history there for you to research.
Something else worth mentioning is that the Falklands come up in Aldous Huxley's "Brave New World", where all the rebel intellectuals get shipped off to remote islands so they don't cause trouble.
Another interesting Atlantic island is
Surtsey, which is near Iceland (possibly too close for your purposes?). In 1963, a fishing ship spotted steam coming from the ocean. That was the first sighting of the new volcanic island. It is now about three square kilometers in size, and has been colonised by plants and animals (but not people yet).
Hope that helps.
