Argentina was a founding member of the League of Nations. However, Argentina left in 1921 when an Argentine resolution was rejected. Why, and through what procedure (if any), was Argentina allowed to rejoin the League in 1933?
It seems that Argentina had become part of the League by International law, not by it's Constitution. The League turned down several of Argentina's proposed amendments (which contradicted other articles of the League's Covenant) at the first assembly of the League of Nations. Argentina promptly withdrew, but never gave a formal notice of withdrawal.
Argentina, in 1933, under a new president, paid its dues and attended several meetings. This was a simpler process beacuse it was officially a suspended or inactive member. Thus, the only things it had to do to rejoin would be to pay its arrears and notify the League of its intent. Paying them off was not very difficult because at the time Argentina was considered a "well-off" nation (it had one of the highest GDP's at the time). I believe it returned mainly because of the new president, Agustín Pedro Justo. He had gained power through the success of a coup. The previous president, Hipólito Yrigoyen, had died the previous year of stomach cancer and the president who had Argentina leave the League of Nations had died earlier that year.
Source: The Origin, Structure & Working of the League of Nations By Charles Howard Ellis