If there is sufficient will for the country to become Independent I think we should allow it to be so, there has been a great dispute between Argentina and Great Britian over who should own the island with neither side agreeing with each other the best possible solution may be for an Independent nation to form.
I do not think the Falkland Islands should become an independent country. Firstly I think that the Islands are way too small to survive on their own; at the very least, they could never defend themselves against outside aggressors such as Argentina. But most of all, the vast majority of Falkland Islanders have democratically expressed their wish to remain part of the UK, so this should be respected.
The simplest answer to this question is that the residents of the Falkland Islands want to remain a British Overseas Territory. The right to self determination should be observed, even if it is unpleasant for the Argentines. The people of the Falkland Islands would prefer to be British citizens than citizens of a small and isolated island nation that cannot defend itself.
The problem with extremely small principalities, such as the Falkland Islands, declaring their independence as nations is that they are not financially capable of sustaining themselves as nations. Being an independent nation requires the installation of a military as well as the construction of an independent government, all of which are very expensive ventures that typically disrupt smaller, unprepared economies.
They country is very tiny and has a small population. It is not large or independent enough to fare on its own. It needs help from a larger country like the UK to keep stable. They should not have been in control from Argentina and Britian needs to keep holding onto the island.