“Freeze a jolly good fellow” is a mondegreen for “For he’s a jolly good fellow.” “THE REFRAIN OF THE ARCTIC CIRCLE—‘Freeze a jolly good fellow’’ was printed in the New York City humor magazine Life on December 4, 1890.
 
A riddle has been cited in print since at least 1997:
 
Q: What do they sing at a snowman’s birthday party?
A: Freeze a jolly good fellow.
 
“Freeze a jolly good fellow” has been printed on several images.
 
     
Wikipedia: For He’s a Jolly Good Fellow
“For He’s a Jolly Good Fellow” is a popular song that is sung to congratulate a person on a significant event, such as a promotion, a birthday, a wedding (or playing a major part in a wedding), a wedding anniversary, the birth of a child, or the winning of a championship sporting event. The melody originates from the French song “Marlbrough s’en va-t-en guerre” (“Marlborough Has Left for the War”). The traditional children’s song “The Bear Went Over the Mountain” is sung to the same tune.
       
Google Books 
4 December 1890, Life (New York, NY), pg. 321, col. 1:
THE REFRAIN OF THE ARCTIC CIRCLE—“Freeze a jolly good fellow.”
 
Chronicling America
20 December 1890, Western Kansas World (WaKeeney, KS), pg. 3, col. 5:
The refrain of the Artic Circle: Freeze a jolly good fellow.
     
20 February 1950, Edwardsville (IL) Intelligencer, “Office Cat” by Junius, pg. 4, col. 6:
He—Have you heard the new Eskimo song?
She—No. What is it?
He—Freeze a jolly good fellow.
 
16 May 1971, Shreveport (LA) Times, “Weekend Chuckles,” pg. 26-F, col. 3:
Mother: Darling, what did you do at the birthday party?”
Little Boy: “Sang the refrigerator song.”
Mother: “Tell me how it goes.”
Little Boy: “Freeze a jolly good fellow.”
 
Google Books
January 1983, Boys’ Life, “Think & Grin,” pg. 66, col. 2:
Jack Frost: Knock, knock.
Old Man Winter: Who’s there?
Jack Frost: Freeze.
Old Man Winter: Freeze, who?
Jack Frost: Freeze a jolly good fellow.—Stan Meier, Alma, Wis.
       
18 December 1997, The Sun (London, UK), “50 best Christmas cracker jokes” by Julian Reed, pg. 8:
What do they sing at a snowman’s birthday party?
Freeze a jolly good fellow.
 
Google Books
The Sleepover Joke Book
By Sandy Ransford
Illustrated by Emily Bannister
London, UK: Puffin Books
2003
Pg. ?:
What do Eskimos sing at a birthday sleepover? ‘
Freeze a jolly good fellow.
 
Twitter
Casey‏
@CaseySisterson
Replying to @jgoode
@jgoode what do penguins sing on a birthday?  Freeze a jolly good fellow.  Okay.  I’m out.
10:03 PM - 16 Apr 2008
 
Twitter
J Å $ Ø И‏
@j9son
- Day 6 of Advent Calendar : Q - What do the reindeer sing to Santa on his Birthday? A - Freeze a jolly good fellow…
6:01 AM - 8 Dec 2008
 
Google Books
Best Ever Classroom Jokes:
Because some of us never grow up
By Mike Haskins
London, UK Pavilion Books
2015
Pg. ?:
What do you sing at a snowman’s birthday party?
Freeze a jolly good fellow.
 
Twitter
Shit Jokes‏
@ShitJokes
What do they sing at a snowman’s birthday party?
Freeze a jolly good fellow.
2:03 PM - 2 Dec 2017