As far as im aware, Argentinas claim to the Falklands isnt even close to being as strong as that of the UK, I believe it was France who first occupied the Islands anyway? The islanders had a vote, thankfully the results of which are being upheld.
It's a bit messier than that, but not much.
The islands were uninhabited and unclaimed.
in 1764 a French ship landed on one of them, founded a teeny tiny colony and claimed the island for the French empire.
In 1765 a British ship landed on a different island, founded a teeny tiny colony and claimed the whole archipelago for the British empire. They didn't know about the teeny tiny French colony on the other island. Neither knew about the other for over a year. Teeny tiny colonies.
A little while later, the Spanish empire claimed the whole archipelago for the Spanish empire because some time around 1500 (I forget exactly when) a pope had decided to divide all land off to the west across the Atlantic ocean between Spain and Portugal. A simple line - everything on this side belongs to Portugal and everyone on that side belongs to Spain. The Falkland islands were on the Spanish side. Not that the pope, the Spanish empire
or anyone else knew that. The Falkland islands weren't discovered until much later. There's some argument about who discovered them. The first person to set foot on the Falkland islands was English, but they might (or might not) have been seen by someone else earlier. But hey, pope.
Some years later, the colonies were abandoned. Britain and Spain declared they would return and their claims were still in place. France didn't.
France never returned.
Spain never returned.
Britain did return and established a permanent and far larger colony.
Note that I haven't mentioned Argentina yet. That's because it didn't exist at the time. The French, British and Spanish claims all predate the entire existence of Argentina. The Argentinian claim is based on a chain of two claims starting with the claim that the French claim became Spanish and continuing with the claim that the Spanish claim became Argentinian and that both claims apply retroactively back to 1764 (when France made their claim) and anyway Spain owned the islands because a pope 500 years ago said so (without even knowing the islands existed) and therefore Argentina does. If Argentina's claim is upheld, that would mean that Argentina owns most of the Americas. Both of them. All the way up to the north pole. Because a pope said so 500 years ago without having a clue what they were talking about.
I'm not kidding about that. The pope's decree is a straight north-south line with no limits. North pole to south pole.