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In French, there's the expression occupez-vous de vos oignons which means "mind your own business" in English but can be literally translated as "take care of your onions".

Know your onions however means to be knowledgeable about a subject - a meaning which is clearly distinct from the French saying.

But what's its origin? I'd really love to know.

This is an American phrase, first recorded in the May 1922 edition of Harpers Magazine:

"Mr. Roberts knows his onions, all right."

According to World Wide Words, this had nothing to do with any Mr. Onions, but:

The crucial fact is that the expression isn’t British
but American, first recorded in the magazine
Harper’s Bazaar in March 1922. It was one of a set
of such phrases, all with the sense of knowing
one’s stuff, or being highly knowledgeable in a
particular field, that circulated in the 1920s. Others were to know one’s oats, to know one’s
oil, to know one’s apples, to know one’s eggs,
and even to know one’s sweet potatoes (which
appeared in a cartoon by T A Dorgan in 1928).
You may notice certain similarities between the
substances mentioned, most being foods and most having names that start with a vowel.

They contain much of the verbal inventiveness
and mildly juvenile wordplay that characterises
another American linguistic fad of the flapper
period, that of describing something excellent of
its kind in terms of an area of an animal’s
anatomy (elephant’s instep, gnat’s elbows and about a hundred others — see my piece on bee’s knees for more).

As with bee’s knees, one of these multifarious
forms eventually triumphed and became a
catchphrase that has survived to the present day.

The Phrase Finder agrees:

Other phrases that refer to
knowing - 'know the ropes', 'doesn't know shit from Shinola' etc. allude to specific items as
the focus of the knowledge.
Other 1920s variants of 'know
your onions' are 'know your oil/
oats/apples' etc. The only one that caught on and
is still in common use is 'know your onions'. So, why onions? Well, as the citation above asks -
why not? Explanations that relate the phrase to
knowledgeable vegetable gardeners, or even to C.
T. or S. G. Onions, are just trying too hard. 1920s
America was a breeding ground for wacky
phrases (see the bee's knees for some examples) and this is just another of those.

Edit: A tantalising snippet in Google Books shows this may have been used in 1908 in a humorous poem in The Postal Record (Volumes 21-22 - Page 27). It' shown in the summary, and is interesting as the year 1908 is also shown. Care must be taken with snippets, as they're sometimes incorrectly dated, but here it is anyway:

But, never mind; Billy knows his onions, He Is not troubled with corns or bunions. He travels along at a good, fair gait; Unless the roads are bad, he Is never late. O. 8. WHITE. WHkesbarre, January 1, 1908. West Hoboken, N. J. At the regular meeting of Branch 1065, held on January 10, 1908, it was honored by the presence of Brother Kelly, President of our National Association.

I find The Phrase Finder's explanation most persuasive:

The English grammarian and lexicographer C. T. (Charles Talbut) Onions
was an editor of the Oxford English Dictionary from 1895 and continued
to write reference works throughout a long and distinguished career.
His last work was The Oxford Dictionary of English Etymology, 1966,
which was published a year after his death. If I knew as much
etymology as he did I could certainly claim to 'know my onions', and
it is tempting to assume that this is where the phrase originated.

If the 'onions' referred to in the phrase is indeed human rather than
vegetable, there is another Mr. Onions that could be our man. S. G.
Onions (they were strong on initials in those days) created sets of
coins which were issued to English schools from 1843 onwards. These
were teaching aids intended to help children learn £.s.d. (pounds,
shillings and pence). They looked similar to real coins and had
inscriptions like '4 Farthings make 1 Penny' or, as in the example
pictured, '12 Pence make 1 shilling'. We can imagine that 'knowing
your Onions' might be coined, so to speak, in those circumstances.

The first known use of 'know your onions' in print, in the 1920s,
tends to argue against either of the above men being involved. While it is true that the phrase originated at a time when C. T.
Onions had established a reputation, the match between the phrase and
his name is just a coincidence. Know your onions is in fact an
American phrase. There are many references to it in print there from
the 1920s onward, but none in the UK or elsewhere until the middle of
the century.

The Phrase Finder goes on to conclude:

1920s America was a breeding ground for wacky phrases (see the bee's
knees) and know your onions is just another of those.

It's worth visiting The Phrase Finder to read the full explanation, which includes a link to other phrases that were coined in the USA.

In the days of wooden walled sailing ships - with no fridges or freezers - onions were a key part of the crew's staple diet. They provided vital protein and vitamins. Moreover, they would keep for a long time provided they were kept dry and in a dark space.

The ship's 'Purser' or 'Pusser' was responsible for storing the ship before sailing and very often he would purchase the necessary victuals from local markets. It was thus vital that the Purser could assess the quality of the food supplies that he was buying - a good Purser would 'know his onions'

I don't know but have always been told that it is was a phrase that started among actors. C.T Onions wrote the first comprehensive glossary of Shakespeare's words. If an actor could speak his part without asking what a word meant he knew his onions. David Crystal has since compiled a more extensive glossary and if I'm working on a Shakespeare play I take it with me. I know my Onions but I don't know all my Crystal.

The phrase 'to know one's onion(s)' first appeared in print at least as early as 1891. Given its historical linguistic context, the 'source' of the phrase may be construed as the independent adoption of a generalized idiomatic response to the equally idiomatic and earlier-evidenced declaration that someone 'does not know the difference between an onion and [another object, commonly another vegetable]'. Both phrases, but particularly the latter, vary freely in individual use, but are loosely formulaic.¹

OED (paywalled) defines 'to know one's onions' as "to be experienced in or knowledgeable about something". The OED entry only mentions the plural variant "onions", attested from 1908 (note the slightly earlier appearance of the attestation shown in the compilation below, from January 1908 as compared to February):

1908 Postal Rec. Feb. 27/3 Never mind: Billy knows his onions. He is not troubled with corns or bunions.

The singular variant "onion" was, however, as far as I have discovered, the only variant in print up to 1908. The singular continued in frequent use into the 1920s, when the plural variant gradually came to dominate use. By the mid-1920s, the plural variant was used almost if not entirely exclusively.²

Early attestation of 'know one's onion(s)'

Early appearances of the phrase 'know one's onions' are a regionalism centered on and predominantly from Pennsylvania. Occasionally, instances from surrounding states (New York, Delaware, and New Jersey) appear, with only one outlier (from Washington State).

All links are paywalled unless otherwise noted.

The Morning Post (Camden, New Jersey) 22 May 1891, Page 1:

Say, Garrett, don't part with Ely, if you know your onion, and we believe you do.

The Evening Journal (Wilmington, Delaware) 06 May 1892, Page 2:

I know my onion, chappies, and doncher forget it — see?

The Morning Post (Camden, New Jersey) 29 Feb 1892, Page 1:

You don't know your onion, Stackey; you oughtn't to pull me in.

The Morning Post (Camden, New Jersey) 05 Apr 1894, Page 1:

This trolley knows her onion and don't you forget it.

The Philadelphia Inquirer (Philadelphia, Pennsylvania) 15 Jun 1898, Page 4:

You must "know your onion" to get your money on right.

The Evening Democrat (Warren, Pennsylvania) 27 Jun 1899, Page 4:

He is a big and busy bug and he knows his own onion and lights on it.

The Seattle Post-Intelligencer (Seattle, Washington)
04 Feb 1900, Page 28:

In cooking onions it is well to first know your onion.

Buffalo Courier (Buffalo, New York) 14 Apr 1900, Page 4:

According to the Binghamton Herald, there is one man in this State who knows 'his onion' as the tough boy says; you can't fool him.

The Selinsgrove Times-Tribune (Selinsgrove, Pennsylvania) 20 May 1903, Page 1:

The people may criticise and damn the old man, but he always knows his onion and rarely makes a mistake.

The New York Times (New York, New York)
07 Jun 1903, Page 6:

"He Knows His Onion."
To the Editor of The New York Times:

Apropos of the discussion of the meaning and origin of "makes no bones," will some slang etymologist let us know the origin and connection of "he knows his onion," meaning he minds his own affairs, or is posted about certain pertinent situstions?

Republican News Item (Laport, Pennsylvania) 25 Jun 1903, Page 1:

The Experimental farmer comes down strong on the butter business, and when he talks about leeks, we have nothing more to say in advice, for it is plain to see that "he knows his onion."

The Index (Pittsburgh, Pennsylvania) 26 Nov 1904, Page 7 (not paywalled):

Not if he knows his onion, and he thinks he does.

Lewisburg Journal (Lewisburg, Pennsylvania) 06 Oct 1905, Page 7:

I know my onion.

Wilkes-Barre Times (Wilkes-Barre, Pennsylvania)
18 May 1907, Page 4 [blurb headline]:

Knows His Onion

The Morning Post (Camden, New Jersey) 26 Oct 1907, Page 11:

It isn't wise for a policeman to say "I never saw the beat." He knows his "onion!" If he made such a remark, in all probability he would not "turn-up" again. He'd get it in the Adam's "apple." That's all!

The Wilkes-Barre Record (Wilkes-Barre, Pennsylvania)
02 Jan 1908, Page 4:

But never mind, Billy knows his onions,

He is not troubled with corns or bunions,

He travels along at a good fair gait;

Unless the roads are bad, he is never late.

  O.S. White, (Letter Carrier No. 9),

  48 Carey Avenue, Wilkes-Barre, Pa.

Early attestation of 'does not know an onion from a ——'

The Scottsboro Citizen (Scottsboro, Alabama) 01 Feb 1883, Page 2:

The Chattanooga Democrat intimates that a man who doesn't know an onion from a grindstone ought not to have been appointed commissioner of agriculture of Tennessee.

Ellsworth Messenger (Ellsworth, Kansas) 15 Apr 1885, Page 1:

If he should take his garden down to his Great Steam Printing Shop, he had better take Jimmy Jarvis along to weed it, for he himself would not know an onion from a bull nettle.

The Gazette (Montreal, Canada) 22 Jun 1888, Page 8:

He did not know an onion from a cabbage.

Woodland Daily Democrat (Woodland, California)
22 Jul 1890, Page 3:

Thad Spaulding and family are there also, and Thad wouldn't know an onion from a potato to-day.

Northumberland County Democrat (Sunbury, Pennsylvania) 19 Sep 1890, Page 2:

Now that fairs are opening we raise to remark that the horse jockey, who does not know an onion from a turnip, generally gets away with more than his share of the boodle.

The Brooklyn Daily Eagle (Brooklyn, New York) 29 Oct 1899, Page 25:

She didn't know onions from lamp wicks and bunions;

A clam from a lobster, or crab from an eel!

Endnotes

1. For purposes of definition, the
   formula underlying the former phrase is "know(s) one's X", and the
   formula of the latter phrase is "do(es) not know the difference
   between X and Y".


2. I do not know why the plural
   variant came to be preferred to the exclusion of the singular. As a
   matter of speculation, I observe that I am slightly more
   uncomfortable with the ill-defined referent of the metaphorical
   "onion" than I am with the equally ill-defined referents of the
   metaphorical "onions". My mild discomfort may be due to what I
   consider the stronger likelihood of inappropriate interpretations of
   "onion" than of "onions".