I use Perlite with the light diffuser on top. I use it because it can be reused from year to year, after it is cleaned and dried out. It does a good job of holding water and does an effective job of keeping things from sloshing when moving an egg box.

Like another poster above, once my eggs start to pip, I move the eggs to a separate container with on a wet paper towel. I will also put a small ceramic bowl of water in the tub, that gets changed frequently.

This method works for me, but there are many ways of doing it. Find the method that you feel most comfortable with and use it. None of them are perfect and almost all of them work.