Rephrased from a previous post, this time serious answers only.
What drove Argentina to claim a set of disputed (even back then) islands that held little to no strategic nor economic value?
It has often been stated that The Islands would have been seeded to Argentina before the Falklands War, and that Margret Thatcher had genuine plans to negotiate the "return" of the Islands, Did Argentina know this and if so why did they still invade?
With regards to why Argentina attacked Britain, it seems most likely that the Argentine Junta, who were in control at the time, were looking for a policy success that would reinforce their precarious political position. The invasion was considered feasible by the Argentinians mainly because the Falklands are so far from the UK and, as you point out, so irrelevant that they did not believe the British would fight. Thatcher's response to the invasion could be similarly considered to have its origins in such political factors, as the successful reinvasion was a huge success for both Thatcher and the Conservative party.
tl;dr: nationalism and political expediency caused the Falklands war
Most of the info I provide here is from Max Hastings book The Battle for the Falklands, so it's only one point of view.
I wouldn't be able to offer an analysis of why the British took the Falklands in the first place however.
It has often been stated that The Islands would have been seeded to Argentina before the Falklands War, and that Margret Thatcher had genuine plans to negotiate the "return" of the Islands, Did Argentina know this and if so why did they still invade?
This podcast - the 1982 one provides some interesting material on what the UK Govt was thinking in the time immediately before the invasion. In short they were negotiating in secret with elements of the Argentine Govt offering the prospect of a "return to Argentina" but with a "pretty much everything apart from the flags stay the same" component to it. This was tentatively agreed between a UK cabinet minister and some Argentine goverment reps but when the deal was floated back in London it was felt that it wouldn't be acceptable to the Falklands population and was put on the back burner.
It's worth saying that similar deals had been kicked around for decades before hand and the fact that one was in play around 1982 doesn't meant that the UK Govt was desperate to close the deal. It might just have been part of a long term "talk until the next regime arrives" type diplomacy.
The wikipedia page covering events before the invasion provides some interesting information about previous British behaviour which might have led elements within the Argentine Govt to believe that the UK Govt would just roll over and accept a fait accompli.
EDIT: to remove idiomatic terms