The Argentine government has made another demand for what they see as the return of their Falkland Islands. The Brits essentially see the Islands as their possession to keep. One cannot return something to another that was never theirs. [David Cameron rejects call to return Falkland Islands to Argentina].
It is difficult to understand why Argentina has continued for centuries to acquire the Falklands. Read the CIA World Factbook or the Wikipedia entry on the Falklands and the conclusion (mine) is that Argentina hasn't any legitimate claim to the Islands.
Maybe more about the ineptness of the Argentina military - the Brits were able to close the distance and easily retake the Islands. The Islands are self governed under the protection of the United Kingdom. And it is this self governance and self-sufficiency, excluding any cost of the British defense, that is, and should be, the final determination.
Why these Islands? The Islands are not full of British expatriates. In fact it is not very full of any human population. They boast of 2,563 people from over 60 nations. The Islands cover 12,173 sq km (4,700 sq miles) of land. The economy is basically fishing for squid and tourism. Given the climate and other factors the Islands sound almost idyllic. See Falkland Islands website.
So why is Argentina so insistent? See this from the CIA World Factbook: "In 1993 the British Geological Survey announced a 200-mile oil exploration zone around the islands, and early seismic surveys suggest substantial reserves capable of producing 500,000 barrels per day."
But there is probably much more than that from the British side. Given the self rule of the Falklands - it is unlikely that oil money would go directly into the coffers of the British government. But here is something to consider - the Argentine government was overly sympathetic to Hitler and Nazi Germany during WWII. Today that sordid past still haunts the Argentine government.
See German Argentine - Wikipedia, the free encyclopedia, Argentina–Germany relations - Wikipedia, the free encyclopedia, Secret files reveal 9,000 Nazi war criminals fled to South America after WWII | Mail Online, Argentina, a Haven for Nazis, Balks at Opening Its Files - New York Times, and The Straight Dope: What's the true story on South American Nazis?
In the end Argentina bluster might be this simple: "Critics suggest that Fernández, an unashamed populist and nationalist, is seeking to deflect attention from social disharmony at home." [Guardian UK].
[Editor: See this from Guardian UK - maybe more info than you need, but good nevertheless: The Falkland Islands: everything you ever wanted to know in data and charts.]