Gravel is unsanitary because the poop gets down into it and can't compost without "brown" organic material to react with.
Additionally, gravel is
forever. If you, or a future owner of the property, ever wanted to do anything else with that part of the yard -- a garden, an orchard, a children's play area, whatever -- you'd have to have heavy equipment mine the gravel back out -- and you wouldn't get it all.
You need to address the source of the water problem.
If the problem is rainwater you need to roof the run -- with gutters to divert the water outside the area -- and add sufficient dry bedding to absorb the water. Aged wood chips are the best for this purpose because they drain freely, absorb a lot of poop, and provide the necessary carbon-rich organic material to compost with the nitrogen-rich poop over time.
If the problem is water running across the ground to enter the run use water diversion methods such as creating grass swales, installing French drains, or even digging open ditches. Some cases even justify using heavy equipment to re-grade the landscape.
If the problem is ground water -- that the water table is close to the surface so that the area is perpetually wet with springs, seeps, and/or standing water -- then this is simply not a suitable place to keep chickens.
If it's the only place physically possible to put them then you need to build up the land with fill dirt in order to raise it well above the water table as well as using water diversion measures to carry the springwater downhill out of the run area.
