While the British already had the Falklands, so were really just retaking what was theirs and repelling an invasion, I think there is more to that. As I've noted before, the British had previously expressed little interest in the Islands in the years prior to the invasion, with James Callaghan (future British Prime Minister) expressing at a Cabinet meeting in 1968 "whether the Falkland Islands were any use to us." They had actively been looking at a means to rid themselves of the Islands, and had been working with the Argentinians throughout the 1970s to hand over sovereignty.

The only issue was that the Islanders themselves had no desire to join Argentina, so this handicapped the British, who were not willing to go against the policy of self-determination (that they ignored this in other regions of the world where they held colonies is another matter). So negotiations stalled on this matter until Thatcher came to power.

In the early 1980s Britain had massive economic troubles, and Thatcher's government became increasingly unpopular; Gallup polls in the fall of 1981 showed her to have the lowest rating ever recorded by Gallup, about 27%. To help bolster the economy massive cuts were made in defence, which included reducing the presence around the Falklands. Conversely Argentina had economic problems as well at the time; only difference is that Galtieri, who took power in a military coup in 1981, invaded to distract from these issues.

Now I would argue that Thatcher took such a strong stance on the Falklands for similar matters. Whereas prior to the invasion it had been said that "on the British side there are probably not more than fifty persons in Britain and in the islands themselves who know what is involved in the dispute of were even, until recently, actively aware of its existence" (this was said in 1969), everyone knew about the plight of the Falklands by 1982.

The speed and overwhelming totality of victory also helped to instill a sense of nationalism in Britain: on her way to Parliament to announce that the Argentine forces had surrendered Thatcher famously said "today has put the Great back into Britain." Approval ratings for her government also soared: she called an early election in 1983 to take advantage of this, and the so-called "Falklands Factor" led to the Conservatives winning the largest majority since the Second World War.

In short, I'd say that Thatcher saw the Falklands Conflict as a means to improve her own standing within the UK. Prior to that she was hugely unpopular, but the conflict turned this around tremendously. It also helped show the world that while the UK was slowly shrinking away as a global empire, it was not just going to be pushed around and allow it to slide back away from the forefront of the world. While it was no longer the massive empire of previous, it was still going to defend its interests.

I think the way you've formulated the question is a bit problematic, because it suggests that Thatcher's desire to recapture the Falklands was, well, a desire. I'm going to argue that it was rather a sense of duty that spurred the response to the Argentinian invasion, and that it had little to do with party politics. Essentially, Thatcher's response was not exceptional at all, and most people agreed with the handling of the situation.

So by the end of March 1982, the Argentine junta had made clear their intentions to invade the Falklands, despite the dissuasion of President Raegan and general global disapproval. Once the Argentines went ahead with the invasion, the UN passed Resolution 502, which essentially demanded immediate cessation of hostilities and the withdrawal of Argentine forces on British territory. However, this doesn't mean exactly what you'd think it means. The resolution was proposed by the British, and they knew that it worked in their favour because it officially recognized the Argentines as the invaders and the British as the victims. Even though Resolution 502 'demanded' a cessation of hostilities, it implicitly allowed the British to react with military force, based on article 51 of the UN charter, which allowed them to claim the right of self-defense given that the Argentines were the invaders. Resolution 502 passed with 10 votes for, 1 against, and 4 countries abstaining.

So basically what I'm trying to say is that the British, and most of the world with them, did not even stop to consider the question of why the British wanted the Falklands. It was a clear violation of British sovereignty and international law, so desire had nothing to do with it. Thatcher's response to the situation, though tinged with her iron-fisted diplomacy, was not considered exceptional. Even if a labour or whig PM was in charge of the country, I would contend that the British would have still responded to the invasion with military force until they recaptured the islands. The response was not borne of a desire, but a duty and an expectation that transcended Thatcher and party politics.

TLDR: the reason Thatcher wanted the Falklands back was that she was operating within a national and even a global system that condoned and even expected a military response to such invasions. She may have had a personal desire to get them back (in fact, I'm 100% sure she did) but the national/international systems were much more important in determining the general response.

As a follow-up question, what odds did the British face in retaking the Falkland Islands? How close was the outcome of the conflict? Did anyone expect Argentina to win?

From the present day, it seems obvious that Britain would have crushed the Argentinians in a conventional war. But judging from many responses in the thread, it wasn't as clear cut. One person even said "Britain could have lost everything". Could they have really?