The Falkland Islands (las Malvinas in Spanish) form an archipelago in the South Atlantic Ocean, off the coast of Argentina. The islands have been occupied historically by the French, the Spanish and the British. However, it must be noted that in 1833 the British Empire occupied the island forcing its Spanish-speaking (Argentinian population) out. That is why today this issue is regarded as a British colonial remnant.

Beyond territorial disputes, the islands have proved to be of geostrategic importance due to its geographic location as a crosspoint between the Atlantic and the Pacific Ocean and as a gate into the Antartic continent. Therefore, the islands have a clear military interest as the country who controls them literally controls access to both oceans.

But more importantly, as many analists point out, the importance of the island is due mainly to the discovery of oil and gas reserves in the islands' soil although it is not yet clear whether its extraction is feasible and economically viable. However, it makes sense to think that both countries, especially England, consider the islands are of high strategic value due to the dwindling oil reserves in the North Sea.

Sources:
peakoil.com/forums/post1080087.html
europeangeostrategy.ideasoneurope.eu/2010/03/07/the-falkland
oilprice.com/Geopolitics/South-America/Argentina-Running-Out