Argentina if she has a claim, has to:
1. Disprove British sovereignty since 1765.
2. Prove that Britain abandoned the Falkland Islands between 1774 and 1833.
3. Prove that the Falkland Islands were part of the Viceroyalty of La Plata.
4. Prove that Spain ceded the Falkland Islands to Argentina (assuming #3 is correct).
5. Prove that Argentina was the sole heir… Because, one must understand that the Viceroyalty also included Uruguay, Paraguay, Bolivia, and parts of Chile and Peru. This brings the problem of “identity of heir”.
6. Prove that the “Convention of Settlement” of 1849/1850 was not a “Peace Treaty”, and that the usual and normal internationally accepted rules and practices of peace treaties do not apply in this case.
7. Disprove Argentina's acquiescence, 1849-1888 and after 1888, in Britain’s possession of the Falklands (i.e. accepted it by action or inaction).
8. Prove that the basic human right of SELF-DETERMINATION does not apply to the Falkland Islanders.
As you can see Argentina has no real claim...