Best, although I don't really know how you'd do it computationally, would be something like a London Subway map... schematically correct but only geographically accurate in a broad sense.
What would be cool about this is that it is still a map of the map, but the unit becomes time adjusted mile.
https://en.wikipedia.org/wiki/Azimuthal_equidistant_projecti...
Perhaps there are metric functions where this is not possible (a quick proof eludes me...[1]), but approximations might be possible anyway.
[1] Edit: actually it's quite simple. Any non-isotropic metric cannot have an Euclidean equidistant projection, simply take a pair of points lying in rays (1,0) and (0,1) in an L^p metric, p not 2, as a counterexample.
[1] http://lazarus.elte.hu/hun/digkonyv/szakdolg/2013-bsc/baranc... (in Hungarian)