I made these coasters with some vintage (old) buttons I've been collecting (estate sales & ebay). I used two different fabrics, one for the front and one for the back. I lined up the buttons on the fabric neatly, marking with a fine tip marker where each hole is, and sewed them in. Then used my sewing machine to sew the fabric pieces together - just like a pillow, leaving a hole to flip the thing inside out. Afterwards, I sewed the hole closed.
Make sure you don't use the type of button that sits up on the fabric, but the type that has holes through the face. Also, make sure the buttons are relatively the same in density. You'll be surprised how flat the coaster ends up being.
Hmmm. I've had this idea in my head for a while and I'm very satisfied with the turnout. To make them even more absorbent, you may want to use a bit of foam between the fabric, but I don't mind. We didn't even have coasters before, so anything is a plus.