I'd suggest sand over gravel. The gravel will help keep the sand from disappearing. The sand is fairly easy to rake out and will drain well if you get rain. Throw some scratch on it and they will keep it pretty well stirred up. I'd think sand may be easy to come by in your area.
If you put organic material in a run, it will tend to rot and make the run muddy and messy. In damp climates it can become moldy. You can compost the organic material but you have to add a lot of nitrogen to it to get it to compost. The chicken manure by itself is not sufficient.
I'd think you don't get a lot of rain so your situation is a little different than most, but I think sand could work out well for you.