What are some physical, geometric, or otherwise useful interpretations of divergent series?

Indeed.

When studying Fourier series, the summation method that is ideal to get uniform convergence of the series to the function is Cesaro summation. See Fejer's theorem.

There are functions that are analytic, for which the Taylor series converges to the function. But there are functions for which this is not true. Using linear summation methods, like Cesaro summation you get the function back. This is a theorem of Carleman that all functions in a quasianalytic Denjoy-Carleman class can be recovered in this way, for certain linear summation method (not only Cesaro's).

You may want to look also to a nice theorem describing all linear summation methods (consistent with the usual summation) by Silverman and Toeplitz.

There are non-linear summation methods too. For example Shanks summation and its generalizations using Pade' approximants.

Notice that the way we assign a sum to a series is, to begin with, already kind of an arbitrary choice. Then, why not consider others.

Series that are classically divergent, and even Cesàro divergent, play an important role in physics.

The canonical example of this is the Casimir effect. When calculating the force of the effect, you are confronted with a divergent series. This series diverges even when you attempt to Cesáro sum it. However, if you use a technique known as zeta function regularization, you can recover a finite and physically meaningful quantity.

There is a lot of hype around about "values" of certain divergent series, like $\sum_{k=1}^\infty (-1)^kk$. But so far nobody has come up with a way of assigning a value to such a series which could be called "universal", or "canonical".

It is, however, true that in some cases a divergent series $\sum_{k=1}^\infty a_k$ can be considered as a limiting case of a family of convergent series $$s(x):=\sum_{k=1}^\infty a_k(x)\qquad(x\in U)\ ,\tag{1}$$ whereby at the same time for all $k$ the limits $$\lim_{x\to\xi} a_k(x)=a_k$$ exist, as well as the limit $\lim_{x\to\xi} s(x)=:\sigma$. One is then tempted to say that $\sum_{k=1}^\infty a_k=\sigma$, but making this conclusion is in fact voodoo mathematics.

Note that the "embedding" of the given series $\sum_{k=1}^\infty a_k$ into an environment $(1)$ involves many arbitrary choices by the master of ceremonies.