Another chance to get myself in hot water
The Falklanders want to remain in the UK. Yet they are not in the UK or Argentina. Let them stand independently and withdraw the military support. They can provide their own and fend for themselves. Soon get fed up of inhabiting a bit of miserable rock in a freezing ocean with no support from the UK. No more subsidies, we can't afford it.
No reason for the UK to think they can hand it over like we own it. The Falklanders can make their own minds up and the Argentinians can stop worrying about the UK being some sort of invading force. If they do attempt to invade, it would be on a sovereign, independent state and other countries can decide if they want to help defend it.
What's more, if there is oil and other wealth and the UK and Argentina want to set up a base, they can bid for it and help the Falklands economy.