As someone who grades undergrad exams involving a lot of elementary game theory I applaud your ambition. If I were in your position, I would start with the prisoner's dilemma, because the unique equilibrium is in dominant strategies and therefore easy to find and understand. However, there is no need to do math in this game, apart from comparing payoffs when deviating or cooperating. But it does illustrate strategic thinking. Contrary to more involved game theory, understanding this game might also have real life benefits. As a professor of mine once said, after you get confronted with the prisoner's dilemma for the first time, you start to see it everywhere.
The easiest game with an equilibrium in mixed strategies is probably matching pennies. Here, you do have to do math to find the strategy probabilities, but I am not sure if this is easily understandable for high school students.
Thus, I would stick to games with equilibria in dominant strategies. See how that goes. If they easily understand it, move on to games with Nash equilibria in pure (but non-dominant) strategies. Only once you have those should you tackle things like mixed strategies and games in extensive form (sequential games), if you want to devote so much time to it.