Try 100% wool felt (maybe a double layer, just in case). Wool absorbs a lot of water before it's wet to the touch. I have knitted wool "coozies" for my ice water in the summer, and they never leave a wet spot.

i made some hexagon ones with cotton batting only. insul-bright and batting should be fine too. just keep them flat.

I have an old shower curtain that cuts and sews like fabric. I cut several layers and use inside and they act as a water barrier. seems to work well.

yes, to me mug rugs are just larger coasters.

I made a set of 18 for my daughter who entertains a lot. I made them as if they were hot pads. Now it doesn't matter if someone sits a hot mug of coffee or a sweating glass of ice tea on them, the good furniture is protected and they all wash beautifully. I used
Insul-brite according to the directions on the package.

I would probably try a layer of vinyl as the bottom layer. I sew with clear vinyl alot for pockets in cosmetic bags, tote bags, etc. and it's never been a problem to sew. Maybe that would keep the moisture from getting on the table.

Hate to be really dumb, but isn't the purpose of a mug rug to be a coaster? A pretty quilted coaster?

"insul.." is the first part of the name. If you're in a fabric store, just ask for the insulating fabric for pot holders.

You can buy sheet vinyl at fabric stores. I put it inside my coasters, where it will not show, and wash them in the machine. I pin all the coasters together to wash them.

I'm glad of this explanation - this Brit has been assuming that a mug rug was an American term for coaster - never heard of it before I saw it here.