Argentina feels that the UK is benefiting from the oil exploration off the coast of the Falkland Islands. And since Argentina feels that they are the rightful owners of the Islands, returning the islands to Argentina would end the struggle. Argentina is fighting back against the UK by imposing new laws about the oil exploration including jailing those responsible for the exploration. It may not improve relations, but it would certainly end the fight.
The Falklands islands have belonged to Britain since Argentina are even their own independent country, it seems unfair that they should be given to them, just because they're closer. This won't improve our relations as it would make us seem that we will bend the knee to any threat that comes near us.
As a colonial power, the British Empire has a history of treating other nations badly. The Empire was built upon land-grabbing, the subjugation of less technologically advanced people, and the violent spread of religion, language, and political ideology. The idea that making a concession of land - over which a recent war was fought - would fix relations between the UK and Argentina/South America is not realistic. Such a gesture would be far too little, too late.