I have a machine that embroiders so I have always used pre-folded diapers and added an embroidery design. You could also applique a design or name on the diaper using scraps. I don't think you can find a material that is as absorbent or that wears as well as a diaper.

I have made a lot of burp cloths for gifts and grandchildren and everyone loves them. I make them with two layers of flannel with warm and natural bat in between. They are in a slight hourglass shape so they fit over the shoulder. The flannel makes them so they don't move around and they are soft. I sew them wrong sides together with a 1/2" seam and then cut away the excess batting in the seam allowance and snip the seam allowance like a rag quilt. A couple of washings and the edge is nice. I can get 2 burp cloths out of a 1/2 yard of fabric. I use the flannel from JoAnn's when it goes on sale. I make one stitching line across the middle to hold the layers together. My daughter uses them to wipe runny noses on her toddlers too as they are so soft.

I make two different ones. The peanut shaped ones are usually 3 layers of flannel. Sometimes I use leftover cotton fabric on one side so it matches the quilt. I really like the pre-folded diaper ones. Just zig-zag a flannel strip on them. Pre-folded diapers are around $13 for 10. I wish I had an embroidery machine because I'd love to embroider on the pre-folds.
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This is what I did, except I did not use any batting. I bought a towel to use as the terry cloth, which might be a bit thicker than terry cloth fabric.

I also made bibs, with a snap closure, from the same fabrics.

My girls like these so well they have asked me to make some for the friends.

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I have an embroidery machine, so I use the prefolded diapers and embroidery a design and put ribbon or fabric to spruce it up. However, I love these pictures of the ones y'all have made. They are so colorful!

I have made many many out of the prefolded diapers. I add a strip on the short edge of the diaper. I also machine embroider or applique initials or something on one end. These make wonderful baby gifts and the new mothers love them and they are super absorbent.

I made some for daughter in different fabrics and then asked her which ones she liked best. I used prefolded diapers, flannel and bar towel. She liked the bar towel backed with cotton best. I prewashed all the fabrics and towels used. I then put wrong side of cotton to towel after cutting off hem and sewed around the edges, leaving a hole for turning. I then sewed the hole closed and around the towel. I then sewed a 2nd line of stitching 1/2 inch from edge. I did add some ribbon on the towel side of some before sewing them wrong side out. She did not like the diapers as they took too long to dry and bunched up in the middle.

These look pretty much like what I've made in the past. I do a little more curving in at the middle area of the burp cloth and I've seen them done with a curve on only one side. Some people say they make them like a doggy bone shape. I've used flannel on most of mine and even a few with terry cloth and cotton. Something that makes good dust cloths when baby has outgrown them.

Thank you so much to everyone for taking the time to answer my question and also post some pictures. I now have lots of ideas and came to realize I can use quite a few of the fabrics I have on hand to make these. I really appreciate everyone's help. THANK YOU!!!

I used flannel with a piece of vinyl in the middle. Absorbent and it doesn't leak through to your clothes