My kids and I have been rewriting old favorites to eliminate references to deities or $mas... most work, if you can get around the rhymes. For instance:

Solstice is coming, the goose is getting fat
Please put a penny in the old man's hat
If you haven't got a penny, a ha'penny will do
If you haven't got a ha'penny then luck to you!

And Deck the Halls doesn't require ANY rewriting, luckily.

We wish you a merry Solstice (x3), which is Happy New Year
We want some figgy pudding (x3) and a cup of good cheer
We won't go until we get some (x3) so bring it right here!
We wish you a merry Solstice (x3), which is Happy New Year!

and, one of my faves we have by Harry Belafonte (which is also unsexist-ed)

I heard the bells on Solstice day, their old familiar carols play
And wild and sweet, the words repeat
Of peace on Earth, goodwill to all.
I thought as now this day I see, the bells of all humanity
Had rung so long the unbroken song
Of peace on Earth, goodwill to all.
But in despair I bowed my head, there is no peace on Earth, I said
For hate is strong, and mocks the song
Of peace on Earth, goodwill to all.
Then pealed the bells more loud and deep
We are not dead nor do we sleep
The wrong shall fail, the right prevail
With peace on Earth, goodwill to all.

I'm going on looking for more...

Sol, a-sol, a solstice, please good missus a solstice
Apple, a pear, a plum, a cherry
Any good thing to make us all merry
One for Peter, two for Paul, three for him who made us all works for me as I have cousins named Peter and Paul so the "him" who made us all is my great-grandfather! But if you have other relatives, you'd have to use their names...