If you don't mind that from the moment you put down a bunch of sand you'll have to perpetually clean the chicken poop from it, go ahead and lay out a bunch of sand. You're also out a bunch of area that could have nice grass and clover that the birds like to eat (plus the grass looks a heck of a lot better than a big sand/gravel pit).
Straw is garbage. It'll never break down in your lifetime (I know, that's a little overboard, but it's not far from the truth!), holds moisture for no one or nothing but will hold the smell of chicken poop real well, and looks like garbage (I'm being extremely nice using the word 'garbage' instead of what I want to call it).
Pallets will work, but they will eventually rot and can potentially be a danger if a slat breaks unexpectedly underfoot and you trip and fall and break something like an arm or leg or worse.
If you can use the pallets, get them (preferably oak if possible), but in about mid-April, put about an inch or two of hay under them, then as soon as it starts to dry up a little, try to get rid of the pallets so the seeds in the hay will get sunlight. This (the hay) will break down well and make the mud into more of a good 'soil'. It will also allow grass to start growing (seeding a yard with chickens around is not easy). If it's not too high-traffic the hay should start things growing quite fast when it starts warming up some. Once you get a good grass established, you won't have to worry so much about the mud problem. The chickens will keep it fertilized so you don't have to buy anything. Get a cheap kind of hay at first, anything just to get something growing quickly (even sage will work in a pinch so *something* will grow fast), then once you have some grass growing maybe find some good hay that, if possible, has rye and bermuda (the rye will grow throughout the winter...well, not 'grow', but will stay green). If you can't get the rye in the hay, get enough rye seed for the area and hand-sow it just before a rain, preferably just after you put the birds up for the night. This way if it rains throughout the night, the seeds will sink down and won't be chicken fodder when you let them out in the morning.
I have a hillside that is nothing but rocks, some cruddy sand and barely any 'soil' (I'm in west TN, btw). When we moved here the very first thing I did was try to cover the hillside to keep it from washing out and making canyons. I used straw because it was late fall and I couldn't find hay for the life of me. This was 4 years ago. The canyons got there anyway, the straw is still there too. Then this year I started to spread the hay on the hillside that I use as the floor covering in my coop. In 4 months the areas I've spread the hay is now growing grass, stopping the washing away and creating a nice soil (the last will actually take a bit longer but the hay is actually breaking down at least, whereas the straw is still there after four years).
If the area that's bad muddy is a high-traffic area, it'll be difficult to fix. If it's at all possible, start another spot with good grass and legume seeds (white clover/ladino clover will make some fantastic root systems that will hold the soil together well and the birds love to munch on it too as it's very good for them) and maybe rebuild the coop in the 'new' area (using the old coops parts of course) once the grass is established well and you can walk the area without it turning to mud. Fence that area off of course so the birds, wild and domestic, don't eat up the seeds, heh. I wish there was an easier 'fix' to muddy spot problems, but there simply aren't any.
