Pigeon-hole Day
Festivities include talking about proofs or arguments that use the pigeon-hole principle (I like that there are lots of these which are accessible). I'm sure there are also games you could come up with (e.g. n+1 people take turns throwing a coin at n jars. If you miss, you go to the back of the line. First one to put the second coin in a jar is eliminated and one jar is removed).
Food: not pigeons please. Maybe a cake with more than n interesting features being cut into n pieces.
One candidate for the day would be every blue moon (whenever there are two full moons in a month), but this only happens once every three years or so. Another idea is to try using the fact that every day of the week occurs 52 times in a year, except one, which occurs 53 times; I'm not sure of the best way to narrow it down from there.