1. Argentina wins war of confederation and absorbs Paraguay, Uruguay, and Riograndese Republics. Have stable if near-monarchical governments provide stability and attract lots of Catholic immigration, maybe Ireland, Spain, and Italy specifically. They can develop with some more time and energy with eventual involvement in Chile and Bolivia over the Pacific War in the 1880s, perhaps taking much of both countries. Improved infrastructure and less infighting could see such a country develop into a second tier power by 1914, especially if there is a successful conflict with Brazil somewhere in there.
In exchange for naval assistance, permission to maintain a naval base/telegraph station, and increased exports of food, Argentina negotiates the transfer of the Falkland Islands along with South Georgia for a considerable sum. Argentina also gets some technology transfer of second-rate UK tech as well, allowing her to advance in many areas. Following the war the Argentineans remain staunch British allies and also eye other territories, eventually conquering Peru as an ally of Ecuador and finishing off Chile. Argentina is careful not to press too hard on her conquests and allows major cities in conquered nations considerable autonomy though Paraguay and Uruguay are ruled as states of Argentina proper. Argentina also builds rail/road infrastructure to unite their nation (much more built than OTL by 1941). UK initially agrees to use Argentina and Brazil as resources in their early war efforts, leading to greatly increased cooperation between the two nations and the beginning of the South American Free Trade Union (UTLSA, or UTraLiSA in local parlance).
Argentina buys British Guyana to secure more land on the continent after the war and also makes Ecuador, Colombia, and Venezuela political.economic satellites over the next 30 years. The fall of the USSR sees Argentina recruit heavily in the devastated country and "Rusiapueblos" appear near major universities throughout the country. Physics, chemistry, agriculture, biology, and materials science beneft tremendously while her military is upgraded with top-of-the-line equipment from old Soviet stocks which is soon improved upon. By 2010, Argentina and Brazil (latter as a junior but distinct power) are in an economic union with most of Central America and the balance of South America.
Both nations are candidates for seats on the Security Council and represent the 5th and 7th largest economies in the world, respectively. Together the combined economic bloc is a competitor to Europe, North America, and Southeastern Asia. A unified currency is not yet present but soon to appear as the Peso Suramericana, and all schools on the continent teach at least Spanish, English, and Portugese. A combined space program has sent men and women into orbit and is preparing to put an independent space station in orbit as a challenge to the independent Chinese effort. They have five aircraft carriers and no official nuclear capacity though it is largely speculated that the Argentineans have up to 200 weapons ready for deployment as the Libertad-D rocket, "reliable as a Proton and as powerful as a Saturn V", can place a significant payload on lunar soil. Coincidentally many of her engineers are Russians, though native-born and mixed-born children are appearing in larger numbers at the facilities.
2. Have King Pedro II of Brazil die at any point early in his reign and *a lot* of the Brazilian rise to power is stopped. The end of slavery might also trigger a civil war that destroys much of the country and could lead to fragmentation, or at least significant decentralization.
3. Combine 1 and 2 for extra punch and maybe Santa Catarina, Mato Grasso, and Acre revolve to Buenos Aires. By 2010 maybe the whole continent is united, though that is a very long shot.
4. Get a stable post-WWII government without Peron that uses its technological infusions for better purposes and develops its resources in the name of the people. Avoid the Falkland Islands War if not the entire military coup altogether. A stable democracy could attract intelgensia from much of the dictatorship-ridden areas of South America and maybe even a few talented individuals from abroad. Should resources permit, have Argentina recruit heavily among Russian scholars to improve her own scientific/industrial base at the fall of the USSR. It would require some luck, but by doing so Argentina could be a match for or even exceed Brazil by 2010.