I can think of several options:
a) put litter in the run (wood chips, dead leaves, anything else that's handy). When it's deep enough, it'll keep the chickens out of the mud, and they can scratch around in it looking for bugs. It will gradually rot away, so you have to keep adding more, but wood chips do last quite a while because they rot so slowly.
b) build another run, and use it while the grass seed is getting started. Unfortunately, you need a really BIG run to have the grass stay alive while the chickens are in there (something like an entire backyard for half a dozen chickens). Swapping back and forth with two runs, you might be able to get grass growing in one about the time the chickens kill it in the other. But you would probably still be replanting it frequently, which doesn't help much with the mud.
c) make a frame that sits on the ground, with wire mesh on top of the frame. The grass grows underneath and and pokes up through the wire mesh. The chickens can walk on the wire and eat the grass that comes through, but they cannot scratch up the roots, so they don't kill the grass. If you do this, you will want to leave part of the run without the wire mesh, so the chickens have a place to scratch and dustbathe and walk on something that's not wire mesh. But you could do a large part of the run that way if you wanted.
Frames like that are often called "grazing frames"--I think if you search, you will find several threads about them.
Sorry, I don't know what specific kinds of grass would be better or worse for your purpose.