Good-quality coarse bark mulch is probably the best option. Also not the cheapest. Straw is cheap and adequate, though it won't last tremendously long. Put a THICK depth of whatever you use in there and it will work better for longer.
Basically any organic matter you put in will decompose and ultimately make the mud much much worse if not removed in time. Finer organic matter (hay, shavings) will rot down more swiftly than coarser (wood chips); the finer stuf is more likely to not get removed in time, too.
DO NOT ADD SAND OR GRAVEL TO AN ALREADY MUDDY RUN. Best case scenario, it helps for a few weeks or months, then disappears without a trace into the mud; usual scenario, it disappears without a trace pretty quickly. Either way you end up wasting all that money and effort with absolutely nothing to show for it (no, having that sand disappear into the mud does NOT change the mud in any discernable way. Illogical but true).
Sand or gravel are a really good solution for a muddy run, of course, but you MUST put them onto fairly bone-dry ground in order to have them last.
The other thing to do -- and you can do it now -- is to minimize the amount of water getting into the run to make mud in the first place. Redirect downspouts; install gutters (yes, even on a small coop!); trench around the run to intercept water and lead it elsewhere; consider a roof or (if weather and structure permit) tarp on the run. The less wet goes in, the less wet the run gets
Good luck,
Pat