If I am right it went like this:
A few french whalers settled on the western island, unaware of the brits at Stanley on the eastern island. The french then sold or gave "the islands" to spain... who in a treaty gave it to Argentina. (It wasn't theirs to give, the British population was apparently about 50x that of the french)
Most notably "Argentina" did not even exist at this time- they were not even a country when the British were already in the Falklands.
Anyway the detail is unimportant. The islands are 300miles away from them.. Can we claim France as British because they are 20 miles away? It's a nonsense
I wonder.. is Argentina's government struggling at the polls again? "Oh we can't control our economy again so lets use the one rallying call we have..."