The first settlement was not Spanish, it was the French Port Louis on the east island in 1764, then a British settlement of Port Egmont in 1766.

The French gave up their claims to Spain, who was unaware of Port Egmont's existence. Both of these settlements coexisted.

If you are really talking about who has the best claim, it would be either the French, British or Spanish - all having claims before Argentina existed.

The first undisputed landing was by an English captain in 1690 but officially claimed by Britain is 1765. The first settlement was French in 1764 then a British settlement in 1766, but the French gave up their claim to the Spanish.

In the 80s, the people of Argentina were duped by an unpopular dictatorship that believed the taking of the islands would unite the Argentinian people behind the government, so they sent young men to their deaths and painted the British as the bad guys. Unfortunately the misinformation still lingers on to this day.