I love the Barenaked Ladies/Sarah Maclachlan medley arrangement of God Rest Ye Merry, Gentlemen and We Three Kings.   I think it could be adapted pretty easily for an instrumental (e.g., piano).

If you're using recorded music, I'd suggest some of the holiday selections by Trans Siberian Orchestra.   There are also some very nice instrumentals on the Chieftains' album "The Bells Of Dublin," and I think you can find them on YouTube too.


Hannukah tunes that will be easily recognizable by anyone who celebrates Hanukkah:
Ocho Kandelikas (Ladino/Sephardic)
O Hanukkah!
My Dreydl
Ma'oz Tzur (Rock Of Ages)
Mi Y Malel
Light One Candle (Peter Yarrow - recorded by PP&M)
Sovivon



Winter songs that have nothing to do with Christmas:
Let It Snow!
Marshmallow World
Winter Wonderland
Frosty The Snowman
Baby, It's Cold Outside
Sleighride
Jingle Bells (Dashing Thru The Snow)
Deck the Hall - (an old Welsh Yule - solstice - song that was later incorporated into Christmas celebrations but is really about the turning of the year)

And, of course, "My Favorite Things" - especially the jazzy syncopated arrangements. This song is often included on Christmas albums but has nothing to do with Christmas either.

"There's No Place Like Home For The Holidays" also is all-inclusive with no reference to Christmas as such