I've been interested in simple paper & pencil games for a while now. Few things are as handy with post-toddler kids as ways to entertain them in a pinch. Here, then, are two games I've found on the net that are riffs on tic-tac-toe.
Now, tic-tac-toe becomes stupid after your first four games. But these games will keep your interest a lot longer:
Abs-trac-toe is a game of occupying space. You apply some simple rules to draw a board with irregularly-shaped regions, and then you take turns marking regions. The player who ends the game with the most adjacent regions wins. That is, you count the borders where two of your regions adjoin to get your score.
Quantum tic-tac-toe was designed as a teaching game about quantum mechanics. You make two moves per turn, but you're marking the possible locations of a move. Only one of those will end up being the actual location of your move, when a measurement is made, later in the game.
Play each of these games a couple of times and you'll have them memorized...and then you'll have two new items in your kidtainment toolkit.