Is there a great mathematical example for a 12-year-old?

Six people at a dinner party is sufficient to ensure that there are either three mutual strangers or three mutual acquaintances. In fact, six is the smallest number that ensures this phenomenon. This is the diagonal Ramsey number $R(3,3)$, and the proof can be demonstrated with a couple pictures and just a dash of the pigeonhole principle. There are lots of directions she could go after understanding $R(3,3)$ (though most of it is not due to Ramsey).

Even though it implies a paradox, I like the Hilbert hotel. It can be explained to everyone, no matter the age. It deals with the concept of infinity, cardinality can be explained in an easy way, if all rooms are occupied and all costumers are in a room, then the "cardinality" is equal. And so on.

I would suggest Euler and his characteristic - for example, use it to show that there are only five regular polyhedra. One advantage of this subject is that one has to only work with pictures and integer numbers.