Given an equilateral triangle with side length 1 and five points within that triangle's interior, some pair of those points is at a distance less than $\frac{1}{2}$. (Or other similar problems using

the pigeonhole principle

.)

Not sure if this following one is one that's commonly seen, but it's possibly a bit more kid friendly: Out of any list of ten integers, there is some nonempty subset whose sum is divisible by 10. (And then, you see you can in fact make it some consecutive nonempty subset of the integers, assuming some ordering to the initial list.)