I have a large stack of flannel rags that my mother made for me. Some are one side old T-shirt and one side flannel, some are double-layer flannel, and some are single-layer flannel. They all do the job. Some of the flannel is cut-up pajama pants or the ends of old pillowcases; most of it is stuff my mom got on clearance at Jo-Ann's. I use them for everything--spills, wiping noses, etc.
They are about 8" by 8" in size or 8" x 10". She went around the edges with a serger, but you can also use a regular sewing machine and just zig-zag. (The edges will fray more using this method, but they'll still work.)
I also have good rags made out of baby drooler bibs--you know, the terry cloth kind. I just cut off the velcro part and zig-zag stitched along the cut part.
I used to keep the pile on top of my microwave, but now they are in a basket on the counter.
If you really want something you can buy ready-to-use, cheap wash cloths would probably work much better than napkins--more absorbent.
Composting is good, but not a perfect solution--trees are still being cut down plus all the nasty chemicals that go into the manufacturing process, the gas to drive the trucks to the store, etc. I do ue paper towels occasionally for greasy jobs, but a roll lasts us for months.