I've been to quite a few of such talks (though most or all in the 12-to-18-year-old range). I feel and believe that a talk that just presents something nice (to a professional mathematician) is unsatisfactory, disappointing, unfulfilling, for such audience.

If at all possible, give a talk that shows the power of math, something with punch: solve a problem. Take a real problem, a problem from reality (*) whose solution is unreachable for the audience, and solve it elegantly with math.

(*) Something you don't need math to explain, to see wherein lies the problem. Euler characteristic, or the hairy ball theorem might get you an "okaaay?, so what?". RSA public-key cryptography counts as reality, by the way (but is perhaps overused).

Some ideas:

• Google's PageRank algorithm might (barely) fit.
• Fractals: are nice. No math punch. Unless you can show, say, that Mandelbrot set represents the set of connected Julia sets. But 9-year-old kids don't get convergence, probably? [I mean, you need to understand at least the definitions of both fractals and of connectivity to feel the punch, the bam!]
• Steiner points in the Steiner tree problem. "Find minimal path network". It's very hard to start thinking about solutions. Unfortunately I don't know Steiner point's derivation, so perhaps it cannot fit in your talk.
• Some other optimization problem, perhaps? Routing?
• On the other hand, presenting a collection of unsolved problems may be interesting, intriguing.

Some comments on your points:

• "Graph theory informs the design of printed circuits." Don't know what "informs" implies here exactly, but my point about punch and 'solve a problem' applies here if you just show that a circuit can be abstracted as a graph ("okaaay?, so what?").
• "mathematics is the only subject that stands on its own". It is not. You can study math on its own, certainly, but it was (and is?) born from reality. E.g. addition for counting sheep, Newton/Leibnitz analysis, and so on. It gives the why.

I've been to quite a few of such talks (though most or all in the 12-to-18-year-old range). I feel and believe that a talk that just presents something nice (to a professional mathematician) is unsatisfactory, disappointing, unfulfilling, for such audience.

If at all possible, give a talk that shows the power of math, something with punch: solve a problem. Take a real problem, a problem from reality (*) whose solution is unreachable for the audience, and solve it elegantly with math.

(*) Something you don't need math to explain, to see wherein lies the problem. Euler characteristic, or the hairy ball theorem might get you an "okaaay?, so what?". RSA public-key cryptography counts as reality, by the way (but is perhaps overused).

Some ideas:

• Google's PageRank algorithm might (barely) fit.

• Fractals: are nice. No math punch. Unless you can show, say, that Mandelbrot set represents the set of connected Julia sets. But 9-year-old kids don't get convergence, probably? [I mean, you need to understand at least the definitions of both fractals and of connectivity to feel the punch, the bam!]

• Steiner points in the Steiner tree problem. "Find minimal path network". It's very hard to start thinking about solutions. Unfortunately I don't know Steiner point's derivation, so perhaps it cannot fit in your talk.

• Some other optimization problem, perhaps? Routing?

• If you talk about chaos (say, in the logistic map, Lorenz atractor or the weather), there's punch in math proving unpredictability [but that's subtle], but the real punch comes if math can say something in spite of chaos and unpredictability (e.g. some general property). [Nothing comes to mind there, sorry.]

• On the other hand, presenting a collection of unsolved problems may be interesting, intriguing.

Some comments on your points:

• "Graph theory informs the design of printed circuits." Don't know what "informs" implies here exactly, but my point about punch and 'solve a problem' applies here if you just show that a circuit can be abstracted as a graph ("okaaay?, so what?").
• "mathematics is the only subject that stands on its own". It is not. You can study math on its own, certainly, but it was (and is?) born from reality. E.g. addition for counting sheep, Newton/Leibnitz analysis, and so on. It gives the why.