I have had 2 pairs of cute socks with giant holes in the heels sitting around for longer than I'd care to admit. I was waiting for the perfect way to upcycle them. It seems that I found it! Little wrist warmers for little wrists.
First, cut above and below the heel and above the toe. You want two straight parts, one from the ankle of the sock and one from the middle of the foot. Work around any holes.
Fold the smaller piece in half so the raw edges line up. Then put it around the larger piece so all three raw edges line up. Sew around. (Serger not required).
Cut a small slit for the thumb towards the end that was the top of the sock. For tiny tots, make it close to the end, because their little hands are so small.
Socks are usually knit material, so it won't unravel.
Legwarmers option: If you use knee high socks and omit the thumb hole, you can use them as infant/toddler leg warmers. Target has lots of cool colors and patterns and they rotate them often. They are usually $2 or less, making them a cheap, yet cute and useful, baby gift.