Adis I agree, but the fact remains, what actual claim do these people have other than many years ago when Britain was a naval power, they occupied these lands and decided to implant British citizens there?
Of course they have certain rights as residents. Through no fault of theirs succeeding generations have developed and built a life there. However it doesn't depart from the fact, that the original claim to the islands is not Britain's to begin with, it was taken by naval force.
Therefore as far their rights go, the decision is not theirs to make, rather through a deal between Buenos Aires and London with proper compensation. If they were an independent country that would be different, they are an overseas territory. The final decision is not theirs but with London or Parliament. The fact that they've been there perhaps centuries doesn't hide the fact they are occupiers and as occupiers they must accept the fact a day may come, when they are asked to leave.

How many people are there? 2000 plus, perhaps 300 families. Ask the Argies to give each family a million dollars or pounds and the British Govt to build or resettle them in the UK with homes. That would be a fairer solution. Then the 2 govts negotiate a final agreement on the islands. The islanders can't always demand their cake and eat it.

And similarly another deal should be done over Gibraltar.