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We say we do not give a hoot or care a hoot when we do not care very much or at all.

On the "hoot" that we do not give, Etymonline has this to say:

[...] Slang sense of "smallest amount or particle" (the hoot you don't give when you don't care) is from 1891.

"A dod blasted ole fool!" answered the captain, who, till now, had been merely an amused on-looker. "Ye know all this rumpus wont do nobuddy a hoot o' good--not a hoot." ["Along Traverse Shores," Traverse City, Michigan, 1891]

Hooter in the same sense is from 1839.

HOOTER. Probably a corruption of iota. Common in New York in such phrases as "I don't care a hooter for him." "This note ain't worth a hooter." [John Russell Bartlett, "Dictionary of Americanisms," 1877]

[Etymonline, "hoot" entry, emphasis mine]

Now "iota", of course, is a plausible metaphor for a smallest amount of particle. Unlike "hooter", which does not seem at all to be self-explanatory as a measurement of anything. (Vaguely, I used to think that I don't give a hoot had to do with hooting as the sound, but admittedly that does not make much sense, either.)

How could "iota" become "hooter"? The explanation above sounds logical, but not quite phono-logical. How could the phonological leap from "iota" to "hooter" take place? Is this a strong case of folk etymology? Was there a sufficient phonological proximity historically?

It's a simple case of someone hearing something he or she never heard before, remembering it wrong, and using the garbled version that actually sticks - some others remember it and begin using it.

Some things are even more intriguing. How does Mary become Molly? How does Margaret become Peggy? How does Ioannes become John, Jean, Giovanni, Ivan, Juan, and, the funniest one of all, Jack?

This is mere conjecture on my part - but based on factual observations.

An alternative form of the word Iota is the word Jot from the Latin Jota, which is a variant spelling of the Greek word Iota. These both refer to the letter i, which it seems was considered the smallest letter in the alphabet, and which by analogy was therefore used to refer to very small things or quantities.

So here is where the conjecture comes in: anyone familiar with Spanish or other languages where 'J' is pronounced as a [x] (the sound often used at the end of the word loch) would be likely to pronounce the word Jota either with a [x] or a [h]. Either of these would make it sound to another English speaker as if it were pronounced "hotta", which might be where we get the variant hooter.

I know this is a mere speculation - but it was too long to put in the comments!

Dictionary coverage of 'hooter' and 'hoot' in the sense of 'minuscule amount'

Etymology Online cites the entry for hooter in John Bartlett, Dictionary of Americanisms, fourth edition (1877), but the entry first appears in the second edition of Dictionary of Americanisms (1859):

HOOTER. Probably a corruption of iota. Common in New York in such phrases as "I don't care a hooter for him," "this note ain't worth a hooter."

It is the truth that politicians who pretend to have such regard for the dear people don't care a hooter, so long as their own selfish ends are attained.—Dow's [Patent] Sermons [1857], Vol. I, p. 6.

John Farmer & William Henley, Slang and Its Analogues Past and Present, volume 3 (1893) endorses Bartlett's etymological analysis of hooter:

HOOTER, subs. ... 3. (American.)—A corruption of 'iota' : e.g., 'I don't care a HOOTER for him.'

Albert Barrère & ‎Charles Leland, A Dictionary of Slang, Jargon & Cant (1889) reaches a similar conclusion, with an additional interesting a potentially very significant observation:

Hooter (American), a comparative for anything worthless or trifling. Bartlett conjectures that it is a corruption of iota, which is also commonly used in New England in a similar manner.

Ah, Billy, you and your sword-cane can't do a hooter among the girls, fine as you think yourself.—Philadelphia Comic Newspaper.

If Barrère & Leland is correct that "not care an iota" was commonly used in New England in the late eighteenth century in a similar manner to "not care a hooter," it seems to me that it offers a strong circumstantial boost to the hooter-as-corrupted-iota argument. Searches of the Elephind newspaper database turn up multiple instances in U.S. newspapers of "not care an iota" (from as early as March 28, 1867), "not give an iota" (from as early as November 25, 1854), and "not worth an iota" (from as early as January 31, 1874). Chronologically, the match is not perfect, since the first "not care a hooter" that Elephind reports is considerably older than the first "not care an iota."

The scholarly J.L. Lighter, Random House Historical Dictionary of American Slang (1997) declines to endorse the "iota" derivation:

hooter n. origin unkn. the least bit; a whit; HOOT. [Earliest cited occurrence:] 1839 in Thornton Amer. Gloss.: Now the Grampus a vessel stopt and didn't buge budge one hooter.

Lighter's mention of whit raises the possibility that hoot might have been influenced by that word (which, Merriam-Webster says, probably comes from wiht, an ancestral form of wight). But here we run up against the problem that "not care a hooter" is considerably older than "not care a hoot."

Christine Ammer, The American Heritage Dictionary of Idioms, second edition (2013) likewise avoids endorsing the "iota" derivation of hooter. Her brief comment on "not give a hoot" appears in the midst of a longer entry for the bundle of similar terms "not give a damn," "not give a fig," "not give a hang," "not give a hoot," "not give a rap," "and not give a shit":

not give a damn Also not give a fig or hang or hoot or rap or shit. Not care about, be indifferent to, [example omitted]. The nouns in all these terms signify something totally worthless. ... hoot has been used for the smallest particle since the later 1800s; ...

It is interesting, therefore, that Merriam-Webster's Eleventh Collegiate Dictionary (2003) doesn't supply a separate etymology for the "iota" definition of hoot as a noun:

hoot n (15c) 1 : a sound of hooting; esp. : the cry of an owl 2 : a minimum amount or degree : the least bit don't give a hoot 3 : something or someone amusing the play is a real hoot

This treatment of definition 2 of hoot as a noun suggests that it, like the owl cry and the amusing thing or person, originated as a descriptive sound. But again, this interpretation does not seem to take into consideration the 1830s instances of hooter as iota, nor does it deal with the chronological evidence that "not care a hooter" seems to be roughly 30 years older than "not care a hoot."

The 1971 full-length edition Oxford English Dictionary is quite disappointing in this regard: it doesn't include a meaning along the lines of "something worthless or trifling" at all. Nor does it include an entry for the phrase "give [or care or worth] a hoot." To judge from its limited coverage of hoot as a noun, the OED is inclined to view the expression "not give a hoot" as being an offshoot of hoot in the sense of "a loud inarticulate exclamation" or of "a shout of disapprobation or obloquy" or of "the cry or call of an owl" or of "apparently a natural utterance of objection or repulsion, there being parallel forms in many languages."

Early appearances of 'hooter' and 'hoot' as 'iota'

Although Etymology Online reports that hooter in the sense of "iota" is "from 1839" (presumably alluding to the example noted in Thornton's American Glossary), I found an earlier instance from an April 25, 1832, letter to the editor of the New York Courier and Enquirer, reprinted in the [Lawrenceburgh] Indiana Palladium (June 23, 1839):

Whin I go to Yaek I hiv to go smack thru Konnettykut—so won Sabbarday i was ridin thru Infield, as still as a kat krawls after a mouse—un just az i got rite aginst the meetin house out popt oald Deekon Parsons, with a s-a-m book in one hand and a tarnal big kane in the tother. Hello! siz he,—yew wikkid kritter don't you kno its Sabbarday? So I put on a pritty midlin kind of a long fase, and told him I was goin to see my ant Nabby—who'd marryd Deekon Amariah Bige-loe. Wal! siz he, Deekon Bige-low or Deekon Bigel-high, u kan't go no furder not'll arter sundown, kaise ime Deekon, un squire—i'm selekt man and kep that tavern, un you must go rite strate thare and stay awl, day; you can get good intertanement fur man un beest.—I told him I waz plagy glad on't for I was pritty darn'd nir half starved un so wus the oald mare and the only reisin I was ridin Sabbarday waz, kaize I had'nt got a single hooter ov munny—but seein he was so good I didn't kare if 1 stade awl nite—so's tu let the oald mare git kinder filled up, iph he had good hay and otes. (I told a most darnashun, awl fired lye—kaize I had my trowsers pocket, stuck chuck full of Arnul's Kimikles.)

This instance offers support for the idea that the term hooter originated in or near New York State as a pronunciation of iota by illiterate or semiliterate persons who may have had no knowledge of the Greek iota.

The expression "don't care a hooter" goes back at least to 1851. From "The Election," in The Carpet-Bag: A Literary Journal (November 22, 1851):

What exciting matters politics are! operating upon the temper of men like the sun of June upon spruce beer, causing them to effervesce, and turn sour, and pop off at the least agitation. We saw much of this at the recent election, where men argued out the merits of respective candidates, and reiterated the fables of the press for six months previous, and all seemed to think they were saying something new : and little men with big argumnts drove big men into little corners, and there run them through with murderous "sillygisms," (Mrs. P[artington]) and laughed at their old jokes as if it were bran new fun. "Gleten" worked well for us : we walked up and, as depicted above from life, like an "independent voter," as we always will be, went the whole unsplit Democratic-Whig-Frcesoil-Scattering ticket, and we don't care a hooter now who knows it. Hooray for our side!

Hoot in the same sense is likewise older than the 1891 instance that Etymology Online cites. From Edward Martin, "Honi Soit Qui Mal Y Pense," in Sly Ballades in Harvard China (1882):

I do not think that for the pelf / Of eligible boobies, / Or for the chance to deck berself / With diamonds and rubies, / Or for her standing in the books / Of prim and proper ladies, / Or for their disapproving looks, / She cared a hoot from Hades.

The phrase "cared a hoot from Hades [or, presumably, Hell]" seems to have been a very early form of "cared a hoot" when it evolved from "cared a hooter." Another euphemistic formulation appears in an untitled item in Life magazine (March 10, 1887):

Here in New York it is believed that at the beginning of Mr. [James Russell] Lowell's remarks, his audience [in Chicago] were not aware whether Shakespeare wrote "Richard III.," or Richard III. wrote Shakespeare. Nor is it believed that they cared a hoot from St. Louis which wrote the other. Most of Chicago's information about Shakespeare comes from Ignatius Donnelly, who is figuring out that he was merely a shadow of Lord Bacon.

An Elephind search for "hoot in Hades" finds 25 unique matches for the expression between May 31, 1905, and October 21, 1973—all but two of them before 1948.

When did popular awareness of 'hoot' as a ruralism for 'iota' disappear

As for when the understanding of "hoot" in these expressions shifted from referring to a very small amount (that is, an iota) to more literally a derisive or owlish sound, the clearest evidence may involve the rise of the expression "give a hoot." Whereas "care a hoot" and "worth a hoot" use hoot in a sense that could easily be understood to mean "a minuscule amount," "give a hoot" makes more intuitive sense when hoot is understood to mean "emit a hoot"—that is, "make a particular sound." (Of course, "give a hoot" could still be understood correctly as meaning "give something of practically no value"—so the "iota" sense of the expression is still not far away.)

Use of "give a hoot" in this literal sense goes back at least to 1873. Consider this extract from a longish poem by Pisistratus Caxton (actually, Edward Bulwer-Lytton) titled "The Boatman," published in Blackwood's Edinburgh Magazine (December 1863), describing a dice game:

Jangle and wrangle, and babel and brawl, / As down from the loud box the dumb dies fall: / A hoot for the loser, a shout for the winner; / He who wins is the saint—he who loses is the sinner.

The poem appears in multiple publications, and is quoted in The Australian Sketcher: With Pen and Pencil (1873) as follows:

The moral to be drawn from which is, that you cannot ask the man you play with in cards, or any other game, where his money comes from. That is his look-out. Yours is to win it if you can. Then let us give—

“A hoot for the loser, a shout for the winner: / He who wins is the saint, he who loses the sinner."

Another early instance of "give a hoot" as make a sound appears in "Hoo-Hoo: As Told at Night by a Hoo-Hoo Owl: The Story of a Robber," in the Camperdown [Victoria] Chronicle (July 28, 1904):

Why, bless you, I'm only four years old, and yet I've hod more adventures than any bird three times my age. Just sit down under this tree and let me give a hoot or two to clear my voice, and I'll tell you a story.

Figurative use of "give a hoot" occurs in Holman Day, "A Yankee 'Horse Shift'," in the [Santa Cruz, California] Evening Sentinel (April 24, 1905):

"Lookin' for a shift?" asks Bangs of Haines. / "I'm pretty nigh suit and I wouldn't give a hoot / "Dunno," said Zeke, layin' down his reins; / For a swap 'less I got some good, big boot."

And similarly, from James Montague, "The Commuter's Dream," in the Salt Lake [City, Utah] Herald (April 14, 1907):

I think, perhaps, I froze to death, / For soon I seemed to see / A Black Man, with a fiery breath, / Who growled, "You come with me!" / "Of course," I answered; "glad to go; / I do not give a hoot / For all your terrors down below— / I commute."

A joke reproduced in The Judge (January 8, 1921) suggests that the connection between hoot and iota had long since vanished from the popular mind:

Not a Hoot—"No, your Honor, he didn't give a hoot whether I saw him coming or not."

"How do you know he didn't give a hoot?"

"Well, he didn't blow his horn."— Louisville Courier-Journal.

Conclusion

Dictionaries have backed off the nineteenth-century authorities' confidence that hooter was a corruption of iota—but they have done so without offering an alternative explanation. The picture is sufficiently imperfect that you can understand why modern authorities are reluctant to affirm the old etymology. But no other explanation makes sense of the chronological priority of hooter over hoot.