As opposed to today's fashion, the focus of Christmas was not receiving gifts. Christmas trees were in use, but mostly in the center of town or a church or other public building, not commonly inside people's homes. The idea of gifts under the tree was not used, though many localities would tie candies and paper gifts to the tree for the children of the town, to be given during a play or other celebration on Christmas Eve or Christmas Day. Gifts in the home were homemade, typically, and not overly large. Usually, they were designed to fit inside stockings (not the over-large ones we use today, but their normal socks, which they washed at night and hung by the fire every night to dry).
So, if the focus was not on the gifts, why would people tell one another "Christmas Gift"? Because this phrase happily reminded everyone that the greatest gift was one that we already had - the gift of Jesus Christ. Jesus came to earth, lived a perfect (sinless) life, died the most horrible, cruel death on a cross, and was raised again three days later. All this was done to pay the price for our sins, which are ugly and change our soul from the image of God into a deformed semblance of the perfection we were made to attain. We now have the chance to wash away that deformity in the precious blood of Jesus through baptism, and then that blood will continually cleanse us as we grow in His Word. This is the true Gift of Christmas, and we need to be reminded on every day, not just December 25, that the price was paid for us. It's up to us to accept this gift and continue to learn how to serve God.
As time went on, this phrase "Christmas Gift" changed, and so did the meaning. Children began using the phrase "Christmas Gift" as a race. Whoever was the first to say it to another was supposed to be rewarded with a gift of some kind from the one who was slower in giving the greeting. In some regions, this phrase changed to one I grew up hearing from my grandmother, "Christmas Eve Gift." Just like the variant of "Christmas Gift", we all tried to be the first to say it so that someone else had to give us a gift. We never actually gave gifts, just enjoyed a moment's superiority over our siblings or other family members. I do not think this is wrong, but it does take away from the original intent of the greeting.
On this happy day when many in the world are celebrating the birth of our Lord and Savior, Jesus Christ, I pray that we remember the true gift of Christmas.
Christmas Gift to You All!