You're talking about a weakly justified seccesionist campaign, with very superficial justification. The inhabitants of the Falklands certainly have more ties to Britain, considering there are no indigenous inhabitants and they're all descendants of British settlers who came before Argentina was a country even.
The British didn't have a continual presence in the islands until 1833, and displaced/absorbed an Argentine settlement founded in 1829 to do so.
Argentina's precursor was established in 1816 and was established on basically its modern borders in 1830.
The waters around the Falklands are of more value than the islands themselves, but one needs the islands to establish a claim to the waters.
And before you bring up the earlier British settlement at Port Egmont, let me remind you that the Spanish evicted them and kept a settlement at Port Louis (Puerto Soledad) going until 1811.
The Argentine claims have more than mere puffery behind them, especially to East Falkland. That said, to put the Falklands under Argentine rule on the basis of them would be ludicrous.
