When I was a kid in the 70s growing up in Brooklyn, we used to use the phrase all the time. The original saying was “Mind Your Big Eyed Business”. So often we would simply say B-Eye business. Not the initial “I” but the word eye. Meaning quit 
in my direction. Hope this helps.
nevermind
So what does the H. stand for in Jesus H. Christ, eh?
Not a phrase that I have seen before. It seems to be used by the Black Community perhaps. I found a quote in a Black Feminist Anthology. It is in a rather sad story about muder and possibly rape (I didn’t read it) called Shoes are Made for Walking. Page 252.
“But Mom,” said Sharon,
“she can’t wear her school shoes to the dance.” Cere’s reply was
swift. “I would thank you to mind your own B I business and
walk up here with me. I don’t have all day to walk home, you
know. Now come on.” The three girls fell silent immediately,
their eyes momentarily focused on her back, and then they sped
up to flank her. Yvette, still pouting, chose the point most distant from her mother.
Given that this is the second reference to NY slang, and it sounds totally plausible, I think this is probably the correct answer.
Hello, Guadere!
Quoth a letter to Cecil, “It stands for Hallmark, because God thought enough to send the very best”.
To which Cecil replied, “I’ll make the jokes around here”.
The H stands for Harold, as in, “Our Father, who art in heaven, Harold be thy name”.
and/or,
Haploid.
Jesus H. Christ, people, can’t you mind your own beeswax?
Isn’t it Mind your B-I-izness ?
There are similar idioms of that era where they spell the start of words.
Apologies if this has already been mentioned.
We lived in a rural area. We said “mind your own bees wax”
I think that’s pretty ubiquitous (in the US) but I may be wrong. I’m in Chicago and I’ve heard beeswax growing up, mainly from older people. I think of it as an old fogey phrase, but still use it playfully sometimes. The (zombie) OP’s phrase is unknown to me.
My mother always said it “B-I-isness” she is from Long Island.
I always assumed it was meant to sound like a failed attempt to spell “business” in a facetious way. Like the character on “In Living Color” who said “It’s a conspiracy, C-O-N-spiracy!”
Another Bronx native here. It was common back in the 50s and 60s. Usually said in a singsong manner: None of your bee eye business! Emphasis on the bee.