I fully expect Argentina to back my claim at the UN, as they seem to think that the following are legitimate grounds for a claim of sovereignty.
A. We went there once.As for the Falkland Islands, there is nothing to negotiate... but this can be settled at the International Court. The United Kingdom is more than happy to meet Argentina there to see if the afore mentioned grounds for their claim trump the Falkland Islander's right to self determination.
B. It's not very far away.
I wish I could claim credit for the idea of settling this in court, but the idea was first put forward by H.M. Government in the 1960s* and as far as I know it is an option that remains open. Alas, Argentina has always ignored this means of settling the matter once and for all.
Correction : This solution was put forward during negotiations in 1940s, 1950s, 1960s, 1970s... until they invaded in 1982.