According to Reader's Digest, "Jingle Bells" was actually not intended as a Christmas carol. If you listen to the words, there is no reference to the holiday or timing other than a mention of "dashing through the snow."
James Pierpont wrote the song for a Thanksgiving program at his church in Boston in 1857. He originally called it "The One-Horse Open Sleigh." The young performers and the audience members enjoyed the song so much that it was repeated at Christmas, and thus began its present association (not associaction) with the holiday.
You can see scans of the original version of the song online at the Library of Congress.
(Image from Wikimedia Commons.)