I'm planning a new coop, permanent, step up from the tractor. It will have both. One roofed run area, for shade and lousy weather. The other will be something added later, just netted over or something. DH is thinking along the lines of moveable tunnels of wire leading to a moveable enclosed ranging area. I've also been looking at the hoop runs based on arched over cattle panels.
Where are you? Climate - wise, I'm looking for summer heat protection and weeks of cold rainy winter mucky.
I've noticed that people have varying opinions on covered runs. I myself have to have a covered run, I prefer it and so does my girls. I live in Oregon, and we get lots of rain and I want my girls to be comfortable. Their first run was not covered and I noticed that whenever it rained they just hung out in the coop area. Now that we have a new coop w/covered run they dont go running for the coop every time it rains, they spend all day in the run now unless their laying eggs, the rain no longer chases them away to the coop.
A solid roof is really nice to have, but does cost more (not just for the roofing, but b/c you need to engineer its rafters etc more solidly).
n your climate, a wire mesh roof would be quite adequate, possibly with the addition of shadecloth on the top and the W side during summer.
Mind, a wire mesh roof still needs to have strong enough rafter-type support that it will not collapse under the weight of raccoons or snow. (Yes, snow CAN still build up on mesh, especially if it is 1" mesh or smaller; err on the side of 'possibly too strong' rather than 'gee I hope that's ok'!!)
Here in Michigan a run is no good 1/2 the year without a covered roof. Where you are at you could probably get away without the roof. Jojufarms hasw a good idea...cover half of your run. THis gives some shade and weather protection, but cuts down on engineering and costs.
Half and half sounds perfect to me, too. We're in Durham, NC and have an 8'x6' section of the run covered with a solid roof. The remainder is covered with 2"x4" welded wire to keep the big predators out. I'm so very happy that we've got a section that stays dry when it rains. It gives me a place to stand around and watch them in inclement weather and I can move the feeder outside and know it will stay dry. I've also located a dust bath for them in this covered section. They really go to town in that thing. It is much more expensive to put up a roof over part of the run, but as far as I'm concerned, it was totally worth it.