Sand or gravel work great to avoid mud... but only if you put them down BEFORE you have mud (or once it has thoroughly dried, in the summer).  As I've said elsewhere a lot of times, if you put sand or gravel into mud, the mud just swallows 'em up. You don't even get sandy or gravelly mud. It is a mystery 
First I would suggest doing whatever you possibly can to minimize water going into the run. This may include putting guttters on the coop or other nearby structures; directing downspouts WELL away from the run (nonperforated black corrugated drainpipe works well for this); roofing the run if feasible; and trenching around it, with an additional arm of the trench to lead water away downhill.
Then, honestly at this point you are probably best off adding coarse organic matter. In the PNW you should be able to get hogfuel; although it varies somewhat it what it's like, it's usually pretty appropriate for this purpose. Or coarsely shredded bark, or very coarse wood chips, or something like that. The coarser the organic matter, the less (and more slowly) it turns into mud itself. 
You will have to rake it all out next spring/summer when things dry out, or your mud will get worse over time, but it should hold you until then (and then you can put sand or gravel in properly and, in conjunction with the things you did to minimize water input to the run, you should be pretty ok)
You are really unlikely to be able to keep white birds completely white in an unroofed wet run in the PNW though 
Good luck,
Pat