We do something similar... but... for adults!
I know what you're thinking, but this can work for adults too! I got addicted to the Cottonelle wipes when they came out. I think 95% of Americans did as well. Those things sell like hotcakes. Anyway, I got sick of paying $7 or $8 for a package - of even the Walmart brand - so I decided to DIY. I figured adult poop is no more disgusting than baby poop, and most of it is gone with plain ol' TP anyways.
I cut cotton flannel (you can use printed flannel) from Joann Fabrics. If you get it on sale it's about $1.50 for a half-yard. I cut them into 4x8" rectangles with pinking shears so it wouldn't fray, and popped them in the same container the Cottonelle wipes came in. Make a mixture of soap to water - I use an old sauerkraut jar and put about 2tbsp of soap in there and fill the rest with water, shake, and moisten the wipes in the container. You can use tea tree oil castile soap if you're concerned with germs. You can use these just like the normal wipes and throw them in your hamper when you're done. Bleach the literal s**t out of them in the wash with your towels and sheets, and you're good to go. The printed cotton flannel hasn't bleached out for me- which is awesome.
No one in my house has ever gotten pink eye, staph, or MRSA. :D