I cut up an old flannel sheet into squares and used a regular sewing machine to do a wide stitch around the edge to keep it from unravelling. They are not the most elegant or beautiful wipes possible, but they do the trick.
It is SUPER soft and seems really nice. I asked the lady at the company what she would suggest for this time of thing and she told me point me in the direction of this fabric. I wanted a fabric that I could cut into squares that wouldn't fray when washed. I didn't want to sew a million little edges. We will see how it holds up after July.
When I used cloth wipes in the past (no longer do with twins..just too much for me right now) I used flannel squares that I simply serged over the cut ends to prevent fraying. I kept them in a large, pretty glass jar with a lid to stay moist. They were actually really cute!