The fabric? It is two layers of cotton with a layer of batting sandwiched in the middle that was pre-quilted. You can get pre-quilted fabric at almost any fabric store. Make sure, though, that the back side is also cotton. There are some pre-quilted fabrics with a creepy poly/nylon backing that would not make for a happy baby: avoid. In addition, make sure the stitching on it is a cotton-poly thread, not a nylon thread. Also unhappy results. Purchase something that has the stitching fairly close together, about 1" squares/diamonds, for clothing. This particular piece I got in Japan. I tell you, those Japanese have some great fabrics! I've been looking around for the pre-quilted fabric like mine to no avail. I did find the cotton print here, but not quilted. Fabric.com also has some nice prints and solids that would be fun to play with.
The stitching? Not done on the serger, or with a double needle. I edge stitched [stitching very closely] down both sides of the bias. I find a point where it is easy to guide the fabric and move my needle position to where I'd like the stitching. Sometimes I will use an edge stitch foot, but I found it wasn't really necessary for this.
The pattern? Not available. Yet. It's a hybrid of a vintage pattern found in my mother-in-law's stash she was downsizing and some changes I made. I will attempt my hand at getting it into PDF format.
If anyone is able to find a source for the fabric I used, let me know!
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Edit: More sources for the non-quilted elephant fabric:
[thank you Sally and Amy!]