What is the accepted method of converting shininess to roughness, and vice versa? Most modern renderers use physically-based materials and their models are often parameterized over roughness. Since this wasn't always the case with renderers, conventional assets often don't have a notion of roughness. Instead, we see "shininess" or "specular power" as a common material parameter.

I understand that there is no exact conversion between the two, but is there a rule-of-thumb / approximate way to get roughness for a material whose specular power or shininess is known?
As you already note, there is no clear cut interpretation/conversion for these values. I think it is even much worse: Depending on your BRDF and internal limitations (like having defined exponents ranging from 2-2048) the interpretation is completely different. Like suggested in the comments, it might be the best to render a series with different values and fit a conversion curve until the value looks intuitive.

A few examples I was able to find some blog posts that mention something about that topic:


At Dontnod entertainment they use a "perceptual linear distribution". 
Sébastien Lagarde acknowledges the problem in this blog post and writes a few notes on that.
Brian Karis uses a squared roughness values in this Microfacet BRDF overview. This illustrates also nicely how differently  roughness is used in different Normal Distribution Functions. Blinn-Phong power is here defined as 2/roughness^4 - 2.
Frostbite uses a squared remapping. ie. Roughness =  (1 − Smoothness)^2
Details about it and their entire material system is explained in section 3.2 of Sebastien Lagarde's writeup.
This blog post suggests to define the roughness for a Beckmann distribution from the roughness alpha as: