I could probably deduce this from the pictures, but what do you value in a rain bike?

Obviously fenders. I think wide-ish tires are important. In the winter the bike lanes here are always chock full of gravel, tree shedding and various other mystery debris. Riding a bike with skinny tires in that muck just isn't fun. I look for tires with a good "wet grip" rating and a fairly minimal tread. Because punctures are more likely in the rain puncture protection is even more important than usual, but I'm still not willing to go the Marathon Plus route. While I know any brake will work, I really like disc brakes for rain riding and especially hydraulic discs.

I'm a bit surprised to see so little IGH representation in this thread. I've always been tempted to build an IGH rain bike, but the weight deters me and I just don't have enough problems with a good old chain and derailleur to make it compelling. I do have a singlespeed that seems like it "should" be a good fit as a rain bike, but frankly while I find singlespeeds to be fun in good weather I feel like having gears reduces the psychological weight of the ride and in bad weather I'm always looking for whatever makes things feel easy. I choose a rack and trunk bag over a backpack in the rain for this same reason.