An old gardening joke has it that one shouldn’t plant onions next to potatoes “because the potatoes will cry their eyes out.” The joke has been cited in print since at least 1912:
 
“We have advised the colonel never to plant onions near potatoes. An Irishman did so once and raised a poor crop of potatoes. The reason, he said, was that ‘the onions made the potatoes cry their eyes out.’”
   
 
7 August 1912, Lowell (MA) Sun, pg. 10, col. 3:
We have advised the colonel never to plant onions near potatoes. An Irishman did so once and raised a poor crop of potatoes. The reason, he said, was that “the onions made the potatoes cry their eyes out.”
(Ed Gallagher of the Concord Patriot—ed.)
 
25 September 1919, Riverside (CA) Daily Press, “Farming a Farm as a Farm Should Be” by Jim Dash, Farm and Orchard Section, pg. 7, col. 4:
A common error among farmers is to plant onions and potatoes in too close proximity. You know what I mean—too near together. The eyes of young potatoes are very tender; and the onions cause them to water so that they just cry their eyes out and never amount to much.
 
4 June 1944, Daily Boston Globe (Boston, MA), pg. 5:
Wouldn’t plant the onions next to the potatoes because he didn’t want the potatoes to cry their eyes out.
   
Google Books
These Were My Hills
By Marie Walston
Valley Forge, PA: Judson Press
1972
Pp. 58-59:
“Why shouldn’t onions and potatoes be planted on the same side of the garden?”
 
Father knew that this idea had come from Uncle Ben, and he had no intention of discrediting the old man. But his blue eyes had a twinkle in them when he said, “I imagine the onions would cause the potatoes to cry their eyes out.”
 
12 August 1976, The Times-Picayune (New Orleans, LA), “Hints from Heloise” by Heloise Cruse, sec. 5, pg. 7, col. 2:
LETTER OF LAUGHTER
Dear Heloise,
Know what I do to get rid of potato eyes?
I put onions with the potatoes, then they cry their eyes out!
HANK
 
Google News Archive
30 July 1998, Daily News (Kingsport, TN), “Grandmothers From All Over Teach Unique Cooking School” by Ellen Wulfhorst (Reuters), pg. 7, col. 1:
“Do you know why farmers don’t plant onions and potatoes together?” she queried.
 
“Because the onions would make the potatoes cry their eyes out,” she said, as her students groaned at the humor.
   
Covered Bridge Gardens
May 15, 2012
May 25, 2012 - Posted by mick
(...) 
Don’t refrigerate potatoes and don’t put them with onions. They will cry their eyes out. Actually the gases the onions give off don’t mix well and cause rotten potatoes.
     
Newton Abbott and District Co-operative Allotment Association
decoydave
04-03-2013, 02:03 PM
(...)
Also, plant potatoes and onions at opposite ends of the garden as ‘the onions will make the taters cry their eyes out’!
 
Coshocton (OH) Tribune
Jul. 3, 2013  
Companion plants can ward off garden pests
Written by Julia Brown
(...)
But remember this old saying, “never plant onions next to your potatoes, or your potatoes will cry their eyes out.”