Ah, now Spain cannot really take political pot shots at us over the Falklands, or even Gibraltar, considering they continue to possess enclaves in Morroco. France and French Guyana. Heck, even Russia and the Kuril Islands.
If the Falkland Islands were not islands, and in fact continguous with the South American landmass, then yes there would be a fair case for Argentina (similar to India and Goa in the 1960s, correct me if I'm wrong).
But we're haggling over relative distance. Distance is not a way to make claims, otherwise Trinidad and Tobago belongs to Venezuela, Alaska to Canada, Kaliningrad to Poland, and China would have no claim to the Spratly Islands.
What difference would it make to Argentina if it did possess the islands? The islands are largely self sufficient, the UK taxpayer barely pays a quid for keeping them and the Falklands don't pay 'tributes' to the UK mainland. In the 1982 invasion the Argentinians immediately enfored the teaching of Spanish in schools and cars driving on the Argentine side of the road. Now, apart from the latter being very dangerous (left-side cars driving on the right isn't fun!), that was coming very close to a form of colonialism, by forcing of one culture and practices over the long established, conquered one.
Taking or gaining the islands would not change anything in Argentina, be it the national wealth, healthcare, crime rates or regional security. Of all things it deprives at least one political party of a unifying point to distract to public away from the proverbial troubles at home.
One last point: should 'Falklands 2' break out, the outcome would not affect UK politics greatly. Foreign policy has hardly ever impacted on British election outcomes. Churchill was dumped in 1945 (Yes! After all of that!); despite Suez 1956 it was another 10 years before they lost power; Thatcher did not survive because of the Falklands, her government's economic plan later in 1982/1983, disunity in the Liberal-Democratic Alliance, and the Labour Party's suicidal manifesto, were far more important factors (Falklands-effect is a retrospective, easy, convenient term); Blair was re-elected after Iraq. Foreign policy is not a political lightning rod over here.
To conclude, Argentina must prove that it can offer the Falkland Islands a better standard of living than they currently have under UK 'rule' (they are largely self-governing if not completely) if they truly wish to possess the islands. But that doesn't change the fact that while they are nearer to Argentina, 300 miles off their coast is still a bloody long way away!