I have a fair solution for this. Just make the islands independent.
There are three types of people on the Falklands. Those who want to remain British, those who wish to be Argentinian and those who wouldn't care either way. To just make it independent would give the islands more trouble than they have right now as they won't really know where they belong in the world. The UK has done a lot for the islands - we have defended them for years, built their roads, hospitals, schools and vital infrastructure. To just give it away would be a huge 'waste' to the UK taxpayer who have been funding our overseas citizens the over years.