To be honest, the falklands are probably the worst place on earth to live, I've been there several times and have no desire to ever return. I can still smell the penguin .... when I think about it, probably mixed with the locals residue from mating with the penguins on an evening due to lack of decent women.

Seriously though, if the locals want to stay there, who are we to argue.

War is always a good way to drum up support when the economy isn't great as a Prime Minister, Thatcher, Blair and Bush all knew that well.
Get the people behind you by showing some muscle and banging the war drums.

I suspect nothing will ever come of it.