What causes deviled eggs to 'weep'?

Assuming it isn't condensation, one thing that causes eggs to weep is the breakdown of the protein structures.

When you cook an egg, the heat causes the proteins to unfold, then refold into different shapes, capturing liquid in the process which is what allows the boiled egg to hold it's structure. These bonds are fairly weak and can be broken down with salt and acids--two things which you have in your deviled eggs.

One thing that might help is slowing down the unfolding/refolding process by putting the eggs in before the water is fully heated and not heating the water up to a full boil. By slowing down the process you are building stronger bonds that won't break as easily. You might also want to experiment with the cooking times, it could be that those times you encountered weeping they weren't cooked long enough to build strong bonds.

I would vote for humidity being a good candidate along with moisture from the eggs.

Also, any covered dish with even slightly warm food put into a cold environment would then create condensation on the underside of the lid. This would then drip down onto your food.

Next time try something absorbant, maybe a tea towel, under the lid for the first couple hours as they cool. Once they are the temperature of the refrigerator they should be good.