Falklands etc would certainly be interesting to Norway during the age of whaling. Maybe the Argentine government saw an opportunity to fill up its sparely populated southern parts during the 19th century, and things just took off?
Argentina was a first world country around 1900. We could either have the country take a different turn from there, or split up during later troubles.
Web estimates say Argentina had a population of about 1 million in 1850, rising to 4 million in 1900. Estimates seem to vary. If just the Norwegian immigration to the US in the period goes to Argentina, it seems Argentiana could very easily be about 50% Norwegian-descended by today.
If nothing else, this would make for a very different political culture and history than the current latin one.
And where did you put the 2 million Italians, 1 million Spanish and 1 million of French, Germans and Poles?
50% is not going to happen. The thing is Norwegians won't settle in the Pampas. Maybe Northern Patagonia, but then there isn't that much space like in the US. Irrigation had to be done.
If things go worse in South America, and the Chile/Argentina area isn't doing well, maybe a more successful Scandinavia could simply buy the place?
Not going to happen. From the 1830s the UK de facto recognized the area was to be partitioned between Argentina and Chile. And thats the only opinion that matters.
Perhaps the Argentine Civil War goes on longer and fractures the nation? Could the independence of Chile be postponed (San Martin never crosses the Andes for example?) so the conflict lasts well into the 1830's while Scandinavia simply grabs Patagonia and begins settlement. How lucrative is fishing and whaling in the area exactly? Could it warrant large scale settlement?
Chile would have gone independent sooner or later. 1825 exagerating.
Besides, the UK wanted no one there, too important for communications.
Patagonia was basically free for the taking until the mid 1800's. There were a couple of tiny settlements here and there, but overall, there was no European population. However, there was a relatively large native population and they discouraged (to put it mildly) any intruders. If you could put together a large enough scandinavian army to drive out (can you say exterminate?) the natives, you've got a large enough army to keep out Chile or Argentina.
The question, though, is why? all but the very northern part of it is worthless for agriculture and mineral extraction (some exceptions). The Chileans/Argentinians only wanted it because the other one did, not because it was seen as any kind of economic asset.
If you can put together your scandinavian army, go for Bahia Blanca and grab the unpopulated (by Europeans) southern Pampas (unpopulated due to the natives) and then work your way south into Patagonia.
I don't want to dissapoint you, but real life is not Europa Universalis. Southern Pampas was officially recognized Argentine by the British. Also it's no land for Scandinavians. Too hot for them.
I will put it this way from my knowledge of the area. After the 1800s, if the UK aprooves, Scandinavia the Southern half for sure. Maybe also up to Neuquen, but thats it. In Chile everything South of Chiloe. Your limit of time is 1833. Or whenever the British try to get the Malvinas/Fk. That made Chile first and then Argentins serious about Patagonia.
Goodlands are plenty. We are too lazy to develop them, they need irrigation, not profitable crops, someone willing to go(like Norwegians).
Now, Tierra del Fuego, inhabited by Scandinavians could very well become an Iceland on steroids.