creaturelife, how big is the run?
 
I would consider putting a swale (shallow channel - I mean just 3 inches deep and 6 inches wide - at the low end of the run, directing the water away from the run, perhaps to a garden or grassy area.
 
Think about a few bags of sawdust pellets - here they are $6 a bag, so maybe $20.  I can get a bale of excellent quality chopped straw for $12 and it covers 100 sf of the day pen.  Not thick, but a nice fresh layer.  Chopped oak leaves might also be good, if you know anyone with one of those shredder vacuum machines.
 
But the first thing I would do is direct the water out of the pen.