Hi Jane,and welcome to the Forum.
I've found it very effective to cover the ground in the run with woven landscaping fabric. You need to either hem the cut edges on a sewing machine, or provide generous tuck-ins at the ends, to prevent it fraying, but if this is done it's very longlasting - mine has been down for 3 years now and is still fine. You peg it down well all round, or else weigh it down with pieces of old broken paving slabs from the recycling centre, and then add a 10cm. layer of either woodchip or rubber chippings, (which is what I use.) The membrane prevents the hens from digging up the mud into the chippings and also the combination of mambrane + chippings provides a warm soft surface for their feet, and something to dig in and bask on as well (unlike concrete, especially in winter when it freezes.) If you then pick up the poo regularly it's easy to keep the run sweet-smelling and comfortable.
As Chris says, some materials do get wet and soggy, but the answer to this is to make at least a part of the run into a dry area, with a roof and also sides which are screened from wind and rain, maybe with a clear or white plastic tarpaulin. These are cheap to buy, easy to fix, last a long time (again, mine have been up round 3 sides of the end of my run in all weathers for 3 years now) and the girls really appreciate shelter from the wind and rain and also filtered shade in hot weather. Hens hate wind, driving rain, and hot sun when they can't get away from it, and do best in a sheltered dry part of their run if this can be provided for them.