Mary, Mara, Miriam, Maria, Mariah or even Merry: "Mary, Did You Know?"
Angelica, Angela, Angelina, Angel: "Angels We Have Heard on High" or "Hark! The Herald Angels Sing"
Belle, Bella, Caroline, Carolina: "Carol of the Bells" or "I Heard the Bells on Christmas Day" or "Silver Bells" (My favorite is the simple, yet beautiful Belle.)
Holly or Ivy: "The Holly and the Ivy"
Nicholas, Nicole, Nicolette: "Jolly Old Saint Nicholas"
Wenceslas: "Good King Wenceslas" (Vaclav in Czech)
Noelle is an obvious choice, I suppose.
Eve might be another ... for a child born on Christmas Eve.
There are a lot already mentioned, so I will have some repeats. Some of the obvious ones from many of these songs are:
Mary, Shepherd, Glory or Gloria, Emmanuel, and Joy. I've listed a few from some of the songs below.
Angels We Have Heard on High - Angel, Angela, Angelica; Meadow (I know it says plains, but I'm taking liberty); Sharon (refers to the plains around Israel); Shepherd
Go Tell It on the Mountain- There are lots of names that mean "mountain", e.g., Montana, Montague, Montgomery, Goran (often used in former Yugoslavia, it means "Mountain main"), Beaumont (means beautiful mountain)
Hark! The Herald Angels Sing - my absolute favorite (mine too!) - Herald/Harold, Mercy, Emmanuel, Virginia
It Came Upon the Midnight Clear - Claire (means clear), Layla (means "night"), Nisha (also means "night")
Joy to the World - Joy (duh), Grace, other names meaning joy, such as Letitia, Kalea (a Hawaiian name), and Rowena
O come, O come, Emmanuel - variations of Emmanuel, such as Immanuel, Manuel, Emmanuelle, Manuela
What Child Is This? - Victory or Victoria, Ransom (kind of weird, but also kind of cool)
Winter Wonderland - Winter, Snow, Jack (you know, because he's nipping at your nose)
The Christmas Song - Merry, Chestnut (wouldn't recommend that one!)
God Rest You Merry, Gentlemen - Comfort
Have Yourself a Merry Little Christmas - Goldie or Golda
It's Beginning to Look a Lot Like Christmas - Silver, Holly, Janice, Jen, Barney, Ben
Mary, Did You Know? - this song gives me chills, I love it 2nd fave song - Mary
Carol of the Bells - Belle or Bella, Dale, Silver
Rockin' Around the Christmas Tree - Holly
Silent Night - Again you could use names meaning "night" such as Layla or Nisha; Franz Xaver Gruber composed the song, so Franz or Xaver would work
O Holy Night - Star or star-related names like Estella or Stella; Hope, Faith