Thanks for visiting us Kayla!
I think in this day and age, anyone would be hard pressed to deny that Christmas has its origins in one or more of several different winter solstice observations. Most of the pagan survivals are Roman in origin. Once the Roman Empire quit prosecuting the Christians and legalized Christianity, the religion was handed over to the Pontifex Maximus. Over the centuries, Christianity took on the favored trappings of paganism, adopting deities as saints and adding their practices to the existing Judeo-Christian teachings and becoming part of the traditions. And the Romans weren’t the only ones who did this.
For Christmas day in particular, the Dies Natalis Sol Invictus (the birthday of the unconquered sun) on December 25th is the most likely choice.
There are several candidates: the Roman sun god Sol, an older sun deity called Elagabalus, or a newer deity with a superficial resemblance to Jesus Christ known as Mithras. Early Christians were very concerned that the Mithraic mysteries would draw their congregation away; although, Mithras' worshippers were primarily soldiers.
Saturnalia led up to Dies Natalis Sol Invictus and originated as the annual celebration of the dedication of the Temple to Saturn on December 17th, but was later expanded to a week-long celebration. A Lord of Misrule was chosen by lot to behave badly and encourage others to turn social conventions on their ear. Gifts were given on the last day of the festival.
Both elements can be found in western European Christmas celebrations; although the Lord of Misrule was repeatedly banned as was Christmas itself on more than one occasion. Christmas was proscribed in England, Germany and America. And remained banned in the United States until 1820.
Why was the holiday prohibited? Because early celebrations of Christmas were notable for their drunken revelry and licentious and larcenous activity. "Men dishonor Christ more in the 12 days of Christmas," wrote the 16th-century clergyman Hugh Latimer, "than in all the 12 months besides."
"A Visit From St. Nicholas" aka "'Twas the Night Before Christmas" published in 1823 was the first reference to the modern Santa Claus. Previously, Saint Nick, also referred to as Sinterklaas, was a thin, strict man dressed in bishop's robes not the fat, jolly elf found in the poem.
Saint Nikolas Day is December 5th, making it a bit of a stretch to link him directly to Christmas Eve. He gave gifts to good children on the eve of his holiday.