I can't find any official source on the Falklands/Malvinas, but Argentina (like most countries) operates a [jus soli] (http://en.wikipedia.org/wiki/Jus_soli) system of nationality, meaning that anybody born in Argentina is automatically a citizen. Argentina considers the Malvinas to be Argentinian soil, therefore any person born there is automatically an Argentinian citizen by birth, even if they don't choose to take it up.

There has only been, as far as I know, one case of a Falkland Islander taking up Argentinian citizenship, as it is understandably considered a political act. That was [James Peck] (http://en.mercopress.com/2011/06/21/falklands-born-james-peck-has-no-plans-to-give-up-his-british-passport), but his situation is complicated because he was married to an Argentinian woman and domiciled in Argentina, so it's not clear exactly how he acquired Argentinian nationality.

You can only really speculate on what the government's practical plans were if it was to gain official sovereignty of the islands. Presumably a number of the locals would have sought asylum in the UK, as they are citizens, and Argentina would have sought to settle Argentinians on the islands.

At the beginning of the invasion, the Argentine Government issued several proclamations stating that the Argentine Constitution applied to all Falkland Residents and providing them information about new government services.

Source: Hime, Douglas. "The 1982 Falklands-Malvinas Case Study". NWC 1036. p. 8.

Follow-up question: How many Argentineans were living on the Islands before the British invasion?

Were the oil and gas reserves nearby known at that time?

Just to add to what others have said because I don't see it mentioned yet but the Argentines did deport a few dozen "trouble makers". The governor, the police chief, a bunch of civil servants and a newspaper editor were deported, some on the first day and some two weeks later. The royal marines stationed there were also deported.