“Rudolph, the Red-Nosed Reindeer” (1949) is a song that was popularized by Gene Autry. “All of the other reindeer used to laugh and call him names” is a line in the song. A joke is that there is a reindeer named Olive, because the song says (based on misheard lyrics), “Olive, the other reindeer.”
     
The joke goes back to 1949, the year the song was released. This is rom the Toronto (ON) Daily Star on December 20, 1949:
   
“A little girl in Huntsville asked in a record store for the record about ‘the two new reindeer.’ The clerk suggested perhaps she wanted the ‘Rudolph’ story. ‘That’s it,’ replied the little girl. ‘It’s Rudolph and Olive; I just forgot their names.’ The clerk then proceeded to tell her the song was only about one reindeer, but his little customer came up with, ‘But it says Olive the other reindeer used to laugh and call him names.’”
   
A book titled Olive, the Other Reindeer (1997) by Vivian Walsh and J. Otto Seibold (illustrator) was published, and this was made into a 1999 American 3D computer-animated Christmas comedy musical film, written by Steve Young. “Who is the meanest reindeer in Santa’s herd? Olive.  You’ve heard the song. ‘Olive, the other reindeer, used to laugh and call him names’” was posted on Twitter by PUNS on December 22, 2022.
     
 
Wikipedia: Rudolph, the Red-Nosed Reindeer (song)
“Rudolph, the Red-Nosed Reindeer” is a song by songwriter Johnny Marks based on the 1939 story Rudolph the Red-Nosed Reindeer published by the Montgomery Ward Company. Gene Autry’s recording hit No. 1 on the U.S. charts the week of Christmas 1949.
 
Genius (lyrics)
Rudolph The Red-Nosed Reindeer
Gene Autry
Track 1 on
Christmas with Gene Autry
Release Date September 1, 1949
   
You know Dasher, and Dancer, and Prancer, and Vixen
Comet, and Cupid, and Donner, and Blitzen
But do you recall
The most famous reindeer of all?
 
Rudolph the red-nosed reindeer
Had a very shiny nose
And if you ever saw it
You would even say it glows
All of the other reindeer
Used to laugh and call him names
They never let poor Rudolph
Join in any reindeer games
   
20 December 1949, Toronto (ON) Daily Star, “Over the Teacups,” pg. 34, col. 2:
Rudolph’s pal ... A little girl in Huntsville asked in a record store for the record about “the two new reindeer.” The clerk suggested perhaps she wanted the “Rudolph” story. “That’s it,” replied the little girl. “It’s Rudolph and Olive; I just forgot their names.” The clerk then proceeded to tell her the song was only about one reindeer, but his little customer came up with, “But it says Olive the other reindeer used to laugh and call him names.”
 
Newspapers.com
22 December 1963, “How Rudolph Came to Christmas” by Robert L. May as told to Alfred Balk, Family Weekly, pg. 7, col. 3:
One writer told how a girl, after hearing the song, resolved never to behave like a reindeer named Olive. Asked to explain, she replied, “I mean, like what she did to Rudolph. You know, ‘Olive the other reindeer used to laugh and call him names.’”
         
OCLC WorldCat record
Olive, the other reindeer
Authors: Michael Christie, David Christie
Print Book, English, ©1988
Publisher: [publisher not identified], Port Coquitlam, B.C., ©1988
   
OCLC WorldCat record
Olive, the other reindeer
Authors: J. Otto Seibold (Author), Vivian Walsh (Other)
Print Book, Undefined, 1997
Publisher: Chronicle Books, San Francisco, 1997
 
Wikipedia: Olive, the Other Reindeer
Olive, the Other Reindeer is a 1999 American 3D computer-animated Christmas comedy musical film written by Steve Young, based on the children’s book by J. Otto Seibold, and directed by Academy Award-nominated animator Steve Moore. The feature was produced by Matt Groening’s The Curiosity Company and animated by DNA Productions.
(...)
The story was based on the 1997 children’s book by Vivian Walsh and J. Otto Seibold and illustrated by J. Otto Seibold. In the song, Rudolph the Red-Nosed Reindeer, the lyric “All of the other reindeer” can be misheard in dialects with the cot–caught merger as the mondegreen “Olive, the other reindeer”.
 
Twitter
Andy Affleck
@aaffleck
Kitchen clean, Ella asking what someone is doing new year’s eve, Olive the Other Reindeer on TV. Time to light the fire place and have wine.
8:22 PM · Nov 22, 2007
     
Twitter
Michael S. Kaplan
@michkap
is warmly remembering Olive (santa’s 10th reindeer). You know, “...Olive the other reindeer used to laugh and call him names…”. #fb
5:21 PM · Nov 20, 2009
 
Twitter
Las Vegas Carl’s Jr
@LasVegascarlsjr
How come u never hear anything about the 10th reindeer named Olive? You know:Olive the other reindeer, who used to laugh and call him names.
10:03 PM · Dec 28, 2010
     
Twitter
PUNS
@ThePunnyWorld
Who is the meanest reindeer in Santa’s herd?
Olive.
You’ve heard the song.
“Olive, the other reindeer, used to laugh and call him names.”
8:47 AM · Dec 22, 2022
 
Twitter
The Dad Joke Man
@DadJokeMan
Santa has an extra reindeer called Olive.
She’s mentioned in the Rudolph song;
“Olive the other reindeer…”
🎄
3:01 AM · Dec 23, 2022