This answer is mostly a follow-up to issues that StoneyB raised (justly, in my opinion) in his answer regarding the claimed connection between the term dido and Dido of the Aeneid.
Webster's American Dictionary of the English Language (1864) has this entry for dido:
Dido, n. pl. didos. A trick; an antic; a caper. To cut a dido, to play a trick:—so called from the trick of Dido, who having bought so much land as a hide could cover, cut it into a long string to inclose more than was intended.
But subsequent Webster's dictionaries backed away from making that connection; and by the time of the First Collegiate (1898), the entry for dido contained the note "Etym. uncertain."
The source of the Dido connection is difficult to pin down. Nineteenth-century periodicals and cyclopedias of knowledge exhibit much the same echo-chamber effect as today's Internet. Google Books returns a dozen or so brief articles published between 1840 and 1880 (many with more than a passing similarity in wording) that link the word dido to Dido in the Aeneid. The oldest one that I could find is this item from an article called "Origin of Words and Phrases," in Samuel G. Goodrich, Robert Merry's Museum (1841):
"He's cut a Dido." It is told in history, that Dido, a queen of Tyre, about eight hundred and seventy years before Christ, fled from that place upon the murder of her husband, and with a colony settled on the northern coast of Africa, where she built Carthage. Being in want of land, she bargained with the natives for as much as she could surround with a bull's hide. Having made the agreement, she cut a bull's hide into fine strings, and tying them together, claimed as much land as she could surround with the long line she had thus made. The natives allowed the cunning queen to have her way, but when anybody played off a sharp trick, they said he has "cut a Dido;"—and the phrase has come down to our day.
However, this account has a serious shortcoming: There are scarcely any instances in Google Books, in the period from 1800 to 1880, of the phrase "cut a dido," except in the word-origin explanations of Goodrich and those who paraphrased him. I found just one such instance prior to 1879 (after which date I stopped looking)—in Susanna Moodie, Life in the Clearings Versus the Bush (1853):
"A dog, sir," quoth the singing master, not in the least abashed by the reproof. "If the brute had cut up such a dido under your bed, you would have been as 'turnal sceared as I was."
On the other hand, Google Books finds numerous instances of the plural forms "cut didos" or (more often) "cut didoes." The earliest instance in Google Books' results appears in Laughton Osborn, Sixty Years of the Life of Jeremy Levis, book 6 (1831):
The Doctor not only appeared to have lost all gaiety of temper, but acted as though his wits were buried in the ruins of his dwelling. He would start at every little noise, like a child that has been reading of ghosts...and press his hands to his head, as though the latter were still aching from the concussion of the earth; and, when asked what ailed him, he would fall to cursing the city, and damning the earthquakes, and swear that, rather than spend another year in Cumana, he would make a second Empedocles, and bury himself at once in the bowels of Aetna. Then, after cutting a few more didos, (if I may apply so vulgar a phrase to a man of the Doctor's refinement,) he would add, more quietly, that his friends must not be surprised if he left the place in a week.
The next instance in Google Books is from the following year, in an uncredited story called "Broker Bullion, or Fashionable Life in Saratoga" in The North American Magazine (December 1832):
and having listened to the consolatory remark of Hayfield that she "need not cut up sich didoes, for every dam would have her day, and some men would marry any thing"—entered the first coach departing...
Various nineteenth-century dictionaries of U.S. slang reported the phrase "cut didoes," starting with John Russell Bartlett, Dictionary of Americanisms (1848):
TO CUT DIDOES. Synonymous with to cut capers, i.e., to be frolicksome. "Who ever heerd them Italian singers recitin' their jabber, showin' their teeth, and cuttin' didoes at a private consart..." —Sam Slick in England , ch. 15. "Watchman! Take that 'ere feller to the watchhouse; he comes here a cutting up his didoes every night." —Pickings from the Picayune, p. 86.
Another interesting comment comes from Maximillian Schele De Vere, Americanisms: The English of the New World (1872):
As long as the rowdy is thus at work in comparative harmlessness, on a spree or in a rumpus, he is very fond of designating his peculiar proceedings as cutting up something, apparently desirous to convey the idea that some mischief, some cutting must be mixed up with it or there would be no fun in it to him. He cuts capers, he cuts up shines, he even cuts didoes, as if he would imitate famous Queen Dido in her cunning device by which she received her magnificent "hide" of land. Such at least is Professor Mahn's interpretation of an expression which so far has baffled all research...."This 'ere Frenchman has been cutting up didoes in my house now for several days; he ain't sober onst a week, and breaks all my cheers and tables, Mr. Recorder." (Pickings from the Picayune, p. 147.)
The "Professor Mahn" mentioned in this extract is "C.A.F. Mahn, of Berlin," whom Merriam-Webster put in charge of etymology research for the 1864 Webster's Dictionary quoted near the beginning of this answer.
Several twentieth-century dictionaries of slang have entries for dido or didoes. Eric Partridge, A Dictionary of Slang and Unconventional English, fifth edition (1961) offers these three entries in addition to the older "cut [or cut up] didoes":
dido. Rum [source]: military (mostly Regular Army): C. 20.
dido. v. 'To steal from carts in the street': Australian cant: C. 20.
act dido. To play the fool: Naval: C. 20. A variant of cut a dido.
Harold Wentworth & Stuart Flaxner, Dictionary of American Slang (1960) reports that "cut didoes" (meaning "to be frolicsome") is obsolete. But it includes this modern entry for dido:
dido n. A complaint; a reprimand. 1958. "Dido—A minor complaint of a superior against a cop." G. Y. Wells, Station House Slang.
And finally, Robert Chapman & Barbara Kipfer, Dictionary of American Slang, third edition (1995) has this intriguing entry:
cut didoes v phr by 1807 To frivol and frolic; =horse around [origin uncertain; the notion of reference to the notorious behavior of the crew of HMS Dido has no confirmation]
I haven't been able to find any information about hijinks aboard the HMS Dido.
It's rather astonishing that the phrase "conniption dido" occurs essentially nowhere on the Internet but in this EL&U question. I wonder if the original poster may have misheard the second word of the term his mother used, or if perhaps she misheard it from her grandmother.
Update (October 3, 2018): Some early newspaper instances of 'cutting didoes'
Although site participant il--ya wins the prize for earliest (as yet) confirmed occurrence of "performed didoes"—dating to 1830—an Elephind newspaper data base search turns up several instances from the middle 1830s. From "A Leaf from a 'Reefer's Log',"in the [Terre Haute, Indiana] Wabash Courier (November 7, 1833):
Up went the pipe of a boatswain's mate, and the shrill prolonged belay, like the continued note of a canary bird, started the whole marine corps and all the officers, on a nine knot laugh. I told the boatswain's mate that if he cut up such a dido again, I would throw his call overboard, and send him to look for It. He grumbled and said,'he bad never been rowed before for piping belay, when an officer gave the word—and if they made a bloody marine of him they ought to take away his call, and clap a stock on him.'
From "The Negro and the Governor," in the [Haverstraw, New York] North River Times (September 19, 1834), reprinted from the New York Transcript:
Some years ago, blackamoor, in Connecticut, returning from a military muster, where he had got practically glorious on an extra glass or two of New-England, was met by the Governor. Both Pompey and His Excellency were on horseback; and as the road was rather narrow, and the negro, in his jolly condition, caused his horse to prance and caracole and cut didos on both sides of the way, the Governor called to him to take one side of the road, and let him pass on the other.
And from "Peregrine Simpkins," in the Richmond [Indiana] Palladium (November 12, 1836):
"Young'uns," remarked a passing Charley, "if you keep a cutting didoes, I must talk to you both like a Dutch uncle. Each of you must disperse; I can't allow no insurrection about the premises. ..."
"Charley" appears to be slang for "night watchman." None of these three early instances of the expression "cutting didoes" hints at any connection between didoes and the legendary Carthaginian queen.