Although a UN referendum on self-determination (one of, it not the, key underpinning principle(s) of these things, don't forget) would be interesting.
Govt says 'right, UN chaps - free and fair referendum, conducted under your auspices to see what the islanders want as per Article 1 of the Covenant on Civil and Political Rights'.
Referendum held - Falklanders vote to remain British. As the thing is UN-run, the Argentines have no chance of claiming that the vote has been rigged - what do they do then? Ignore self-determination principle and try to invade? Sanctions against the UK on the grounds of... what, exactly? Upholding self-determination?
It would be a potentially useful mechanism to wrong-foot the Argentines (perhaps coupled with rumours that a couple of SSNs are in the area, just in case).
And it'd save us multiple threads on here about how a carrier is the one and only way of protecting the Falklands (Pprune, passim)...