If we're going by proximity, there's some Caribbean nations that are even closer to the US than the Falklands are to Argentina, would you argue that we should annex Cuba or the Bahamas?

And from what I understand, the people of the Falklands overwhelming want to be a British territory. I think that's probably the more important consideration.

It is wild that it came to be the way it is. It certainly doesn't make sense to me in the world before modern air travel, the internet, etc. that they'd be ruled by a country so far away, but in this modern era where just about anywhere in the world is only about a day's travel time, or available on-demand 24/7 by phone or computer, it makes every bit as much sense to me that they be a UK territory as it does that Alaska is a US state.

Hawaii is actually a pretty interesting comparison to make, because most Hawaiians did not want to become a US territory at the time, but that's really begging a whole 'nother discussion with lots of complex talking points about imperialism/colonialism, indigenous rights, etc. but I'm frankly just not going to go into that right now. Suffice it to say that it's similar in the sense of it being a small island territory located far from the colonial power that laid claim to it, but the attitudes of the people living there were very different.

I'm no historian or anything of the sort, so take my thoughts on this for what it's worth (and I am certainly biased being an American, don't exactly get a whole lot of Argentinian history books to study, and most of the Spanish I know is food-related, so if someone wants to enlighten me more on the Argentinian side of things, I welcome the education.) But in general my understanding is that the British were the first people to land there, didn't really do much with it at that time, and pretty much just said "finders keepers"

Maybe worth noting, there were no indigenous inhabitants there, so that's probably about as ethical as colonization can get.

Then France showed up and set up shop since the British weren't doing anything with it. Britain came back and also set up shop, and it's not totally clear if either of them even knew the other was there. France eventually decided to fuck off, and let Spain have their bit of the Falklands.

Spain and Britain coexisted for a while, had some scuffles, but more or less worked things out. Eventually Britain pulled out to focus on other things but still considered their "finders keepors" claim to be valid.

Spain eventually pulled out as well, so for a little while no one was really doing much of anything with it officially.

Argentina (technically Buenos Aires at the time if we want to split hairs, I'm going to just use Argentina and Britain to keep the sides easy to follow) comes along, and decides it's theirs, and this is pretty much the root of the dispute. While Britain still held their claim of "finders keepers" Argentina countered with "losers weepers"

Argentina gave some German dude permission to set up a colony for them there to fish and hunt feral cows. Eventually he gets into a fight with an American navy captain over fishing and hunting rights, Captain America kicks their ass a bit and declares the colonial government disolved, and pretty much continues on his merry way. Argentina tries to get things there started back up again but never quite gets their shit back together in the Falklands. A little while later the Brits come back around, still claiming finders keepers, and take charge of everything again, and this time the colonies stick and continue to grow. Argentina spends the next hundred years or so muttering "this is bullshit" to themselves.

Around the 1960s, Britain starts talking about decolonizing, and Argentina gets excited thinking they're going to finally get the Falklands. Britain even quietly floats the idea of giving them the islands, figuring the Islanders would just kind of accept that decision if it was made, and running these islands from halfway around the world was getting kind of expensive. Turns out though that pretty much everyone on the Falklands is pretty damn happy to be British subjects and don't really want to be part of Argentina, which made things a bit complicated.

Argentina gets kind of impatient with all of this, and eventually decided "fuck it, we'll just take them ourselves," Britain cannot abide Argentina's inability to wait patiently in the queue and was starting to really wrap their heads around the idea that the Falklands would rather stay part of Britain and so we get the Falklands war.

Britain wins, Argentina goes back to muttering to themselves, and that pretty much brings us up to the present day.