We know about it's failings, what about some of the good things it did?
The World Heath Organisation, which is a subset of the United Nations, owes its beginnings to the League of Nations. They had some very real and marked success with work throughout the world with vaccinations and education programs. They also carried out a number of assessments and made attempts to formalise healthcare in countries such as china.
A lot of their efforts, especially those in China fell foul of the failings of the league as a whole, and especially activity in China was limited after Japanese protests and then the subsequent invasion.
This is a nice little read that touches on the sorts of activities, and makes some assessments as to their success. I will try and find some more detailed sources too.
EDIT: I'm struggling to find anything of any real substance freely available on the internet, and as my university recently ended its support to alumni for Jstore I'm afraid I'm a tad out of luck on that front too. Hopefully what I have already posted is suitable enough.
One event that is often held up as a success is the "Åland Crisis" of (1918-)1921.
Åland is an island situated between Sweden and Finland. When Russia took Finland from the Swedes in 1809, they also acquired Åland. However, the population of Åland remained Swedish (in 1921 as many as 90% of the population of the island considered themselves to be Swedes.)
In 1917, following the Russian Revolution, Finland declared its independence from Russia. The Ålanders, concerned that being a part of Finland would lead to the "Finnification" of the islands and the suppression of their distinctly Swedish culture, declared their own independence and sought to be incorporated into Sweden.
The Swedes and Finns agreed to send the matter to the League for a decision on whether the island should remain a part of Finland or become a part of Sweden. In the end the League decided the island belonged to Finland (despite 90% of the population being Swedish and wanting to be a part if Sweden).
The Swedes abided by the Leagues decision, and conflict (which had seemed possible when the Swedish Navy arrived on Åland to evacuate civilians and then did not leave) was averted.
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It to some degree dealt with the Armenian refugee crisis. Although they were unable to secure them a national homeland in the republic of Turkey or in the Armenian SSR, they were abel to relocate them. The city of Aleppo in the French mandate a.k.a Syria was designated as a resettlement place for the Armenians.