Wolf Moon
The usual explanation is that wolves howled with hunger outside villages in the depths of winter and the month's moon took their name from it. It is a good image and the behaviour is wrong: wolves howl to assemble a pack and hold territory, and they do it year round rather than out of hunger. What is true is that January is the middle of the wolf breeding season in the northern hemisphere, when howling does increase — so there is a real thing behind the story, just not the one the story tells.
Where the name comes from
The name reaches most people through the Old Farmer's Almanac, which attributes its list to Algonquian peoples. That attribution is doing more work than it can carry: the published lists disagree with each other, several of the twelve are plainly colonial English rather than Indigenous, the almanac's own sourcing is thin, and the nations grouped under 'Algonquian' span a continent's worth of distinct languages and calendars — many of which counted moons rather than months and did not map onto January at all. 'Wolf Moon' is a real and widely used name today. Presenting it as a single traditional one is the part that is not supportable.
Also called Old Moon, Moon After Yule.
Old Farmer's Almanac full moon names; sourcing discussed rather than assumed