"This board is the only time in my life I have ever seen Ave Maria associated with Christmas. In my view there is no connection."
It's been included on a ton of Christmas albums, from Crosby to STreisand. But that's not what makes it appropriate. The Words come from the New Testament story of the Annunciation, in which the angel Gabriel appears to Mary to tell her she is with child. (Mary, not yet wed to Joseph and still a virgin, is dismayed.) But the angel explains that she has been chosen to be the mother of the Messiah, and without any inconvenient sex act required. Bam, she's pregnant.
The Annunciation is celebrated on March 25 - 9 months before Christmas, appropriately.
So, the Ave Maria is related to Christmas in that the prayer to Mary begins with the words she heard when the angel told her she was to be a mom on what would later be known as Christmas Day.
BTW, On Dec 8 Catholics celebrate the feast of the Immaculate Conception, a somewhat related day: Mary's birth is celebrated Sept 8, and Dec 8 recognizes her own conception (9 months earlier) as the only human conceived without original sin, post-Adam and Eve.