Chariklo,
Thank you for that. While I understand what you are saying, most long standing conflicts such as this have multiple stories and angles at which to view them. The British has had an administation since the 1830's you are correct. Argentina claims that the Spainish held it prior to the british, and that they received it when they became independant. There is confusion as to wether the Portugese, English, French, or Spanish "found" it first in the 16 century. The spanish and english both arrived and left, claiming it as they went (as as was the custom of annexing land at the time).
Since we live in a more civilized time, and it makes sense that the island 300 miles off the coast of Argentina should belong to it, I would say it would be the right thing to work with Argentina on a mutually beneficial arrangement if they are willing. The nationalism of Argentina may not allow them to do this, and the UK does not HAVE TO do a damn thing because the resident want to be British. They have access to GB, europe, and a social safety net. If they became Argentina, they would have none of that. So no duh that they want to be British. :)
It belongs to the UK. End of story. But they can do the "right thing" and share the resources in a way beneficial for all, without releasing their claim to the land.
Recognizing annexing land claims on the other side of the world made centuries past is kind of beyond the 21st century. This is why the UK needs to find a solution if Argentina is agreeable. It is entirely possible the Spanish landed first. Their plaque still stands, as does the english one I believe.