“Mayonnaise” sounds like “mayo neighs.” A group of singers were called the “Mayo-Neighs” according to a 1968 newspaper. A horse was named “May O’Neighs” in 1975.
 
A riddle was published in a 1997 riddle book:
 
Q: What do horses put on their salad?
A: Mayo-neighs.
   
A joke was posted on Reddit—3amjokes on January 15, 2016:
 
“I have a horse named Mayo. Mayo neighs.”
 
     
Wikipedia: Mayonnaise
Mayonnaise (/ˈmeɪəneɪz/, /ˌmeɪəˈneɪz/, also US: /ˈmæneɪz/, and often abbreviated as mayo /ˈmeɪoʊ/) is a thick, creamy dressing often used as a condiment. It is a stable emulsion of oil, egg yolk, and either vinegar or lemon juice, with many options for embellishment with other herbs and spices. Proteins and lecithin in the egg yolk serve as emulsifiers in both mayonnaise and hollandaise sauce. Commercial egg-free alternatives are available for vegans and others who want to avoid animal products and cholesterol, or who are allergic to eggs. Mayonnaise varies in color, but is often white, cream, or pale yellow. It may range in texture from that of light cream to a thick gel.
             
15 June 1968, The Evening Capital (Annapolis, MD), “Mayo Players In Top Form,” pg 21, col. 5:
The Mayo-Neighs then let loose with “Cincinnati, Ohio,” providing a nice change-of-pace with a bit of square dancing.
 
4 November 1975, Wellsville (NY) Daily Reporter, “Batavia entries,” pg. 9, col. 1:
May O Neighs (RMays)
(The name of a horse.—ed.)
 
16 April 1977, Syracuse (NY) Post-Standard, “Buffalo Entries,” pg. 15, col. 1:
May O’Neighs, Mays
(The name of a horse.—ed.)
 
29 April 1985, The Irish Times (Dublin, Ireland), “Letters to the Editor: The Origin of Mayonnaise,” pg. 11, col. 5:
The word however, does have a strong Irish connection. A tribe inhabiting the West of Ireland discovered a piquant form of horseradish sauce which had a peculiar effect on the vocal chords. Hence the name—a corruption of Mayo-neighs.—Yours, etc.—
DES MacHALE,
24 Avonlea Court,
Blackrock,
Cork.
     
Google Books
Best Riddle Book Ever
By Charles Keller
New York, NY: Sterling Publishing Company, Inc.
1997
Pg. 14:
What do horses put on their salad?
Mayo-neighs.
 
November 1998, Ranger Rick, “Reader riddles,” pg. 35:
What do horses like on their hamburgers? Mayo-neighs.
Erin Beecham; Warrenton, OR
     
Google Groups: alt.humor.puns
Meanwhile at Mayo
Tom Nelson
1/12/00
The Mayo Clinic has opened a veterinary branch for horses. Now Mayo neighs.
 
Twitter
Dré Labre‏
@drelabre
Q: What condiment goes best with Horse? A: Mayo-neighs.
2:32 PM - 22 Feb 2010
   
Twitter
Oly Grocery Outlet‏
@OlympiaGO
Bad Joke of the Day:  What do horses put on their sandwiches?  Mayo-neighs!
2:52 PM - 15 Aug 2010
     
Twitter
Marty‏
@martymcfries
I have a horse named #Mayo. He #neighs
1:25 PM - 27 Jul 2011
 
Twitter
BulletsAndBBQ‏
@BulletsAndBBQ
Replying to @_s2_11
@stoo11 a midnight sandwich with horseradish mayo neighs
11:20 AM - 2 Dec 2011
     
Google Books
Beware Of Smiling Faces:
A Short Story
By D. Michael Werts
Published by author
2013
Pg. 16:
“What- what did the farmer tell the city man w-when he asked why he named his horse Mayo?” She took another step down and could not clearly see the rope that tied Evan to the railing. He didn’t answer between his chuckles which became more guttural by the moment. She was almost at the bottom. He looked so sad with the smile on his face.
 
“Because Mayo neighs.”
 
Twitter
no i’m Wendy‏
@ninadiazzz
Wouldn’t it be cool to have a horse named mayo.So, when some one says “Did he just neigh?” You can say “Yes, Mayo neighs” like mayonaise lol
12:11 AM - 22 Feb 2015
   
Twitter
CJ Marx‏
@CjMarx8
How does a horse named Mayo talk?
Mayo neighs.
10:48 PM - 1 Mar 2015
 
Reddit—3amjokes
I have a horse named mayo… self.3amjokes
Submitted January 25, 2016 by wifebeetr6969mayo the horse
Mayo neighs.
 
Reddit - Jokes
What is a horse’s favorite condiment?
submitted February 17, 2016 by lemonwisp
MayoNEIGHS
     
Facebook
Lame Jokes
October 21, 2016 ·
I have a horse named mayo… : Mayo neighs.
 
Reddit—Jokes
I have a horse named mayo…
submitted November 19, 2016 by madazzahatter
Mayo neighs.
 
Reddit—Dad Jokes
I bought a horse (self.dadjokes)
submitted July 8, 2017 by Ezeei
I bought a horse and called it mayo because mayo neighs.
 
Twitter
LaffGaff‏
@LaffGaff
I have a horse named Mayo.
Mayo neighs.
#funny #jokes #jokeoftheday
4:12 PM - 19 Feb 2018