i've always loved cloth napkins. over the decades i've bought some in stores and luckily found some cheap at the church flea market and some at garage sales. dont think i ever paid full retail price for them tho.
some are the linen a bit like tone on tone fabrics but the texture is real not just visual. they also sell them at places like cost plus or pier one imports if you have those near you. prices pretty good too as they last for years and years.
i love them. using them everyday. they are so much more absorbant than paper napkins. tho i dont know about using regular quilting cotton, i dont think it would be heavy enough.
i'd say either look for them on sale everytime you're out and about or buy some yardage of the linen tablecloth fabric, cut and sew the edges down and should work a treat.
mine vary in size from 10inch square, not as useful to about 16 inch square. you could also have a good nosey at the ones in the shops to see what you think is better size to make your own.
...and keep a close eye on those garage sales etc around you. amazing how many folks now cant be bothered adding to the laundry, they're so small doesnt make a lot of difference to me, so they sell cheap with what they think is just more stuff...we all know better tho, eh. i just hated using paper and half way thru a drippy meal needing to get up and get more napkins.
hope that helps some.