As nearly as I can tell from the pictures, this looks like simply row upon row of either 3 single crochets or 3 double crochets (depending on how high the stitch is) all done in the same stitch below.
Once you've done the 3 stitches, they fan out and give some ease so that the piece doesn't curl up like a shrimp! Because of that, you can then go onto the next set of three stitches without putting any chain stitches in between.
Each set of 3 is worked in a single stitch below. I can't tell exactly which one but you work the 3 stitches either between the groups of 3 stitches below, in the center of the 3-stitch group or it looks like it could be in the first stitch of the 3 stitch group. You'd have to take a close look at it to determine it for sure.
The edging appears to be two rows worked around the outside edge of the finished center- the first round would be to attache your yarn, make a longish loop, say chain 5, skip 3 stitches along the edge, single crochet in the next stitch and continue around. Then on the next round fill the loops with single or double crochets, maybe 7 of them to create the pretty loop effect.
As far as the yarn, it might be a good idea to take the "blankie", with daughter attached naturally, to a yarn store or even a Michaels or AC Moore (or whatever is in you neighborhood) and try to compare. It's most likely a baby soft yarn of some type. Can you reach the person who made it? That could give you some answers.
Meanwhile, use whatever you can to tie off the stitches that are unravelling. Even if it has a hole or two, I'm sure you little girl will still love it!