A solid roof on the run is always an excellent idea, gives you the most secure (predatorproof) and weatherproof and dry run possible. You will not be depriving chickens of the experience of being rained on, trust me, rain will blow in from the sides
HOWEVER, the cost of roofing a run adds up real quickly (especially if snow and/or windloads have to be considered). Basically you are building a wall-less shed (in fact, you're probably best off using pole-type shed plans, but will need to add diagonal bracing to the walls due to the lack of rigid siding). The cost of the posts, beams, rafters and the metal roofing itself kind of adds up.
This past winter I built one of what, when the ground dries, will become two roofed runs, 6x14 and 10x10, and the framing/roofing materials for the two of them together will be almost $1000. (This excludes wire mesh for the sides, which I am using recycled chainlink, but most of the lumber was bought new, as was the Suntuf roofing, only slightly more expensive than metal would have been)
So, before committing to anything, add it all up. If you can swing it, though, you really won't regret it -- it makes managment EVER so much easier. Mud type problems especially. Also it's easy to make very chicken-friendly for wintertime.
Good luck, have fun,
Pat