This question came from our 5 year old granddaughter. I had to admit I'd forgotten and that we'd need to look it up on the Internet.
The term goes way back to Anglo Saxon times at least. Long before the welfare state and charities as we know them today, folk used to place gifts in a box for distribution to the poor the day after Christmas Day. So "unboxing" day would be more accurate.
Victorian employers made it more popular by distributing the Christmas Box (accumulated tips) to staff on that day.