As newborns are still developing control over their hands, it's very easy and common for them to accidentally scratch themselves, most noticeably in the face. Newborn mittens help prevent scratching, but the store bought ones tend to come with the worst kind of elastic that allows babies to easily pull them off.
Industrious mom Brittany L. got tired of her little guy slapping his mittens off and instead made her own mittens using one sleeve from an old t-shirt and KAM snaps! To make her mittens, she folded one end of the strap a few times and sewed it together for reinforcement for the snap. On the end that gets sewn into the mitten, she cut some little squares and sewed them on for reinforcement, leaving enough fabric beside the reinforced area to sew the strap into the mitten.
You can also use socks, and some fold over elastic with snap fasteners for straps.