Being a member of the Jordan family meant hanging one's stocking on New Year's Eve, for that was how the Jordans did it. So--if memory serves correctly -- did the Rooses and the Ehrats and a few other families in the small midwestern town where I grew up.
As children we considered it very odd indeed to hang a stocking on Christmas Eve. Stockings, as everyone knew, were filled by the brownies who slid through the mail slot or slipped under the door while children who had been allowed to stay up till quarter past midnight slept heavily.
Not much else was known about brownies (thus necessitating no elaborate lies on the part of parents). Once we became more sophisticated in the traditions of other families we felt a little superior and extraordinarily lucky to have New Year's stockings. My children still do. The Case for New Year's Stockings
* They are in keeping with psychologists' advice on spreading the holiday out to keep children (to say nothing of adults) in one piece. There's less day-after-Christmas crash because part of the festivities is still ahead.
* Parents have a second chance. If the item they overlooked turns out to be the anticipated gift, a watch (or certificate for a soccer ball) can still be tucked into the New Year's stocking.
* One can sometimes get another pay check under one's belt.
* Stocking-stuffers can take advantage of post-Christmas sales and uncrowded stores and have more leisure to poke around for things that delight.
* The small items are better appreciated when they don't compete with the litter of wrapping paper and glitter of bicycles on Christmas Morning.
* New Year's--basically an adult day -- now has some interest for kids as well.
* Kids being left home with a sitter while the adults go partying feel life has some compensations.
* Children are occupied while their parents sleep in on New Year's morning and have new diversions for those last restless days before school starts again.
* The void you anticipate (probably needlessly) can lead to a new tradition of popcorn-making or family charades. By New Year's morning the kids will be converted.