The UK doesn't really get anything out of owning the Falklands. It is very much a net loss financially, but, they weren't going to let the 1800 residents be subjugated by the Argentine military junta that had a record of killing its own citizens.
Before the war, the UK government tried not to antagonize the Argentines by limiting exploitation of the islands' natural resources, so the only economic activity was sheep farming. But, after the war, the gloves were off, and now they have a thriving fishing industry, and are about to start drilling for oil offshore (which is ironic, because the UK government won't exploit its own oil reserves at home). However, instead of this money going to Westminster, to pay for defense of the Falklands, they are letting the local government keep the money, which it intends to use to start a sovereign wealth fund.
If Argentina continues making progress towards being a stable, capitalist county, and the UK keeps heading in the opposite direction, i could see ther being a point at which the islanders would accept some kind of limited Argentine sovereignty, as a face saving way to placate the Argentines, and ensure that Westminster doesn't completely sell them out, as the leftists in the UK seem likely to do.
