i'm sick of the bloody rain, too!
yes, sand or gravel will definitely help eliminate mud, because those two substrates don't turn into mud, even with the constant stinking rain we've been having in the northeast. if money is no object, i'd recommend a three to five inch layer of very coarse gravel or crushed stone with two to three inches of sand on top of that. you can tweak that recipe a bit to fit your budget. your goal here is to put as much well-draining substrate as you can over your non-draining clay soil. resist the urge to put down any kind of organic material, such as straw or shavings. that's going to decompose and make a layer of extremely water retentive compost, which will exacerbate your mud problem. also, hay and straw will mold fast in standing water and you don't want your birds running around in moldy muck.
good luck. here's hoping for a nice long drought. *sigh*