I am no security expert, but for what it's worth, I use a pretty strong base password, which is a couple characters in the middle which vary based on the name of the account. The base password is multiple permutations of some very personal information. So even if forgot my passwords, I could probably figure them out eventually.
A fantastic book that helped put me on the path to math grad school was William Dunham's "Journey Through Genius". Every chapter builds up to an important proof, explaining the historical context and necessary mathematical ideas along the way. By the end of the book, it hits the idea of different kinds of "infinity", and why there are "more" real numbers than integers! That's a really deep, fun and important theorem! I really can't recommend this book highly enough for a motivated high-schooler.
To show a student a completely different kind of math than they're used to, you might want to let them check out some topology. "The Knot Book" by Colin Adams is a tractable introduction to knot theory, which isn't a bad way to get some exposure to topology.
Polymer physicists are into chains.
