I know how to nondimensionalize a PDE with constant coefficients. For example, if you have a simple diffusion equation on $\Omega \in \mathbb{R}^2$ \begin{equation} \partial_t u = D \Delta u . \end{equation} You introduce the dimensionless variables $\overline{x}= \dfrac{x}{L}$ and $\overline{t} = \dfrac{ t}{\tau}$ where $L$ and $\tau$ are length and time scales. Then, with a little bit of abuse of notation, we get the dimensionless equation \begin{equation} \partial_{t} c = \dfrac{D \tau}{L^2} \Delta c. \end{equation} Now the parameter $\dfrac{D \tau}{L^2}$ is also non-dimensional. In more complex equations, we divide by some parameters and make choices of $L$ and $\tau$ to reduce the number of parameters in the equation.
However, what if we have some variable parameters, for example, say the reaction-diffusion equation \begin{equation} \partial_t u - \nabla \cdot (D(x) \nabla u) + b(x) u = f(x). \end{equation} Where $D, b,$ and $f$ are all variable parameters. How do you nondimensionalize this equation?
You can nondimensionalize using 'characteristic' values (constants) of the variable diffusivities, etc, that carry the units of each variable. For example, consider $D_0$, $b_0$, $f_0$ to be characteristic values of $D$, $b$, and $f$, then nondimensionalizing would result in $$ \frac{d\bar u}{d\bar t} = \left[\frac{D_0\tau}{L^2}\right]\nabla\cdot(\bar D(\bar x)\nabla\bar u) + [b_0\tau] \bar b(\bar x) \bar u = \left[\frac{\tau f_0}{u_0}\right] \bar f(\bar x) $$ where $\bar D(\bar x) = D(x)/D_0$, $\bar b(\bar x) = b(x)/b_0$, $\bar f(\bar x) = f(x)/f_0$ are all dimensionless functions. I've also included the scale for $u$ here ($\bar u = u/u_0$).
By nondimensionalizing with respect to a characteristic value you will have set some property of the dimensionless functions. For example, if $D_0$ is defined to be the average value of $D$ over the interval, then the average value of $\bar D$ is $1$. If instead $D_0$ is the value at a boundary, then $\bar D$ at the boundary will be $1$.
As you say you could then choose $\tau$ to set one of these non dimensional groups to one, and then the others will be non dimensional parameters measuring the relative importance of the other terms (if you can choose $L$, and/or $u_0$ you can set the other groups to $1$, but it might make more sense to set these by initial/boundary conditions and domain size, depending on your problem).