Wow! I have missed quite a bit. I love that the argument of generations long since past that many would prefer to forget, somehow yields relevence to current conflicts.
I honestly believe it is simpler than that. The islands are a hop skip and a jump from Argentina. They really seem to belong to Argentina from a reasonable standpoint. The residents however wish to remain British and that cannot be ignored. The dubious claims of various nations on the islands throughout the years aside, a compromise would be fairly simple. Like most things in Argentinas history, when things are bad, you whip up the national fervor in an attempt to distract. It is exactly what happened in 1982, where both Argentinians and British both died.
So now, witht he current (and constant) economic issues creating a grumbling majority, the nationalism is whipped up about the islands in order to distract.
Britain could easily defuse this and do the "right thing", by drawing up a treaty of sorts that allows a shared revenue and cost sharing of the mineral and oil exploration. Argentina could give full support, the islands would remain British, Argentinians could freely visit and use it as a territory, and they could carve out special excemptions.
If the Argentinian President refuses any sort of compromise, than too bad. The international support would rest squarely with Great Britain (except for Ecuador, Peru, Bolivia, and Venezuela), and they could keep the status quo.
It simply cannot be overlooked that reasonably, the islands are related tot he Argentinian mainland. Its not like California would allow the Santa Cruz islands to be part of Mexico if there were enough Mexicans living on them for them to want to be part of mexico.