I think some background will make the kind of answer I'm looking for clearer. I'm trying to think of an elementary proof of the Pythagorean Theorem. I don't like the geometric proofs because they all seem to rely on some kind of trick or clever construction. In particular, they don't seem to shed any light on why we look at the square root of the sum of squares to find a distance.
By contrast, the inner product seems like an elementary way to prove the Pythagorean theorem that does give us some understanding of why we should square and take square roots. Briefly: you use Gram-Schmidt to convert a basis to an orthonormal basis. Taking the inner product of a vector with itself in the orthonormal basis immediately gives you the Pythagorean theorem.
This seems like a promising approach since the defining properties of an inner product (linearity, symmetry, and positive definitenes) are pretty basic. But, even though they're basic, I'm not sure how to motivate them geometrically. So, at the level of geometry, why should we want to consider an object like an inner product on a vector space? Of course, we know by Cauchy-Schwarz that the inner product helps us define magnitude and angle, but I don't think that's obvious from its defining properties.
Perhaps the thing to do is to start with the relationship $\langle v, w \rangle = ||v || ||w|| \cos \theta$, where $\theta$ is the angle between $w$ and $v$, and then argue that $\langle \cdot, \cdot \rangle$ has the properties of an inner product.