Whats the connection here in this phrase? I saw someone say a black guy should eat a mayo sandwich to turn white, in a sarcastic manner. I have seen this several places and it doesn’t seem to make much sense unless it is the color white?
Probably a reference to the movie Undercover Brother from 2002:
http://www.metacafe.com/watch/an-EJKeJJY4tht7m/undercover_brother_2002_eating_mayonnaise/
It’s been a stereotype at least as far back as the 60s.
The stereotype doesn’t apply to all white people, just WASPs.
I suppose an annexure to the OP’s question, if one might be permitted - is it actually a thing? As in, “Are there really significant numbers of white people who eat mayonnaise sandwiches?”
I’ve certainly never met anyone who does, but I grew up in a country where butter/margarine/olive spread are almost de rigueur for any respectable sandwich and mayonnaise was mostly something for salads.
I never really believed it was a real thing, posting at the wee hours of Sunday morning doesn’t lead to eloquent threads though. 
Depending on the base filling, mayonnaise is a fine condiment. I cannot picture a chicken sandwich without it.
However, as a practicing American White Person for several decades, I have never encountered a member of my group who had ever made a sandwich of only mayo the way a person might make a bead-and-butter.
Your Mileage May Vary, and I cannot speak for a great number of sub-groups such as “Crackers,” “WASPs,” “NASCAR Fans,” “Trust-Fund Babies,” “Theatre People,” or any family traditions which are not Celtic or Eastern European.
White man with poor dietary habits here, and I’ve never done it or heard of anyone else doing it. Is the point that mayo on white bread makes for a completely white sandwich, at least if you cut the crusts off?
So, it is likely to be a out-group “joking” insult, similar to the Mexican barbecue problems.
White mayo on white bread (cut diagonally) on a white paper plate eaten by a pale nerdy man wearing polyester plaids.
That’s the image that immediately popped into my head. I know not from where it commeth, but I do recall a neighbour who ate white bread and american cheese sammiches with gobs of mayo on them. He was a socially inept engineer (not on a train), so maybe I am mixing a couple of memories.
This probably goes back at least to the 1950s and maybe much earlier. The difference back then wasn’t so much white vs. black, but WASP white vs. ethnic.
Here’s a snippet from The New Yorker, from 1941:
There was a real divide in New York, the center of Yiddish culture (what most people call Jewish culture, but it’s really one particular type of Eastern European culture), whose rye bread and pumpernickel sandwiches were famous and were always covered with mustard instead of mayo. All the other ethnics in New York also had their particular styles of breads, usually whole grain or seeded or otherwise different. Supermarkets did not. Those were also the days of hamburg sandwiches and hot dog sandwiches - not served on buns but standardly on white bread.
By the 1950s, when the non-ethnics moved to suburbs, moms spreading mayo on white bread came to symbolize “American” culture - moms slapping a couple of pieces of packaged baloney (not bologna) onto bread and sending the kids off to school with it. The use of mayo on everything was a yucky constant - somebody is going to bring up James Lileks’ Gallery of Regrettable Food so it might as well be me.
And if you had this ready made-symbol of white on white to apply to white culture when the back-white divide loomed larger than the ethnic-white divide, how could anyone resist?
I’ve made Miracle Whip sandwiches, in the throes of poverty. Mayo is just too dull on its own.
My mom used to eat mayonnaise sandwiches as a kid, and I’ve seen her eat them a handfull of times over the years. She calls them “sammy sandwiches.” I’m sure I ate one or two as a kid.
We are white, BTW.
In the spoiler below is a link to the Urban Dictionary’s entry on “mayonnaise sandwich.” Hardly an authoritative source, I know, but still…
I’m spoilering the link because the second definition for “mayonnaise sandwich” is definitely NSFW. 
Use of mayo is simply a stereotpye for suburban WASPs, has been for some time. Im reminded of scene in Northern Exposure when Jewish Joel Fleishman from NYC visits WASPy Maggie OConnels family in Grosse Point MI. At a party buffet he finds potato, macaroni, chicken, and tuna salads, along with sandwiches dressed in mayo. He wonders out loud, “is there anything you people eat without mayonaisse?” It is a thing that comes up when people play with stereotypes, but ive never met anyone who eats sandwiches made of mayo and bread only.
I’m so white I make Queen Elizabeth look Third World, and the very *thought *of mayo makes me gag.
Mayo sandwiches without peanut butter are just wrong.
My mother used to give me lettuce and mayo sandwiches for lunch on occasion, though.
Add ketchup. The sweet and tanginess of the ketchup goes great with the creaminess of the mayonnaise. The umami and saltiness help a bit, too.
What ethnicity that would make me, I don’t know.
Mom made Miracle Whip and Cheddar sandwiches when I was a kid. Always on wheat bread, though. She never allowed me to eat white bread. I wouldn’t have thought to eat it without the cheese. Someone from Texas offered me a bite of her peanut butter and mayo sandwich once. I tried not to gag as I swallowed it down and said, “interesting.” Sooo not for me.