Why did Argentina seize South Georgia immediately prior to the Falklands invasion?

The purpose was to help prop up the tottering government by a foreign adventure so that the populace would forget about the poor results in economics and internal development.

By taking all the bases in the region it would just make it more difficult to respond, and why would any sane country bother over some sheepfarmers on a set of worthless rocks?

But Britain managed to mount a counter-operation (with some US aid) and not easily or cheaply retook the islands and restored the status quo ante bellum. One thing the US did was lend them use of our facilities on Ascension Island to provide a base for long range aircraft.

To bargain and negotiate in a strong position. They didn't expect a war, but only a diplomatic conflict. So it wasn't a military strategy. In the worst case the expected to give away the other islands (South Georgia and Sandwich) while keeping Falklands.

I completely agree with you that attacking South Georgia was a strategic error.

What you have to realize is that the people making these decisions are humans, not robots, and often the reason for an action has more to do with politics than what is militarily optimal.

Since South Georgia was more or less unoccupied, the Argentinian leaders could immediately announce a success by invading it and their would be no countervailing factors; no civilian deaths, no occupation issues, no chance of resistance. It would appear to be a completely "safe" victory. It is hard to resist the lure of safe victory, even if it has no strategic value.

The main reason as I understand it was to extend the Argentine territorial waters and thus create a stronger claim to ownership of the Falklands. It would also seriously deplete the British forces in the Falklands to oppose the invasion (though that was of course not guaranteed).
Of course there's also the potential wealth of oil, gas, and minerals in and around the islands, but those were not as important back then as they would be today (the cost of extracting them was relatively much higher because of the lower prices of oil and minerals in the 1980s, making extraction at the time not economical).

South Georgia was invaded in early March, on the 19th the British noticed an Argentine flag flying over the islands, well away from their own scientific station there. The Argentinians had landed armed forces disguised as scrap metal dealers on the islands.
The next day HMS Endurance, embarking part of the Royal Marine detachment to the Falklands, sailed from Port Stanley to confront the Argentine landing force at South Georgia. On the 24th Endurance reached South Georgia, but was ordered to reinforce the British outpost there rather than sail around the island to the Argentine camp, possibly because of uncertainty about Argentine strength in the area.
This was fortuitious, as that same day the Argentinians landed another roughly 100 troops on the island (Endurance carried only 20 Marines and was itself only lightly armed, more a police vessel than a warship).

Directly following this, the Argentine junta found itself pretty much forced by general Anaya (who had orchestrated the South Georgia affair seemingly on his own) to bring forward the invasion plans for the Falklands. The decision to land in early April was made on 23 March, initially the plans had been for a landing about half a year later.

This according to "The Falklands War" by Martin Middlebrook, a military historian who's studied the whole affair in depth (as much as possible, the Argentinians still often don't want to talk to British people about it all).