Stall pellets will start to disintegrate and decompose almost instantly -- in an already muddy run they will make the problem worse over time, even if replaced weekly. (the more organic-y your soil gets, the more of a sponge it is) Coarse shavings would be better, tho still have those problems...
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...