The phrase accidents will happen in the best-regulated families, and variants, are elaborations on accidents will happen, and variants, meaning accidents will happen despite efforts taken to prevent them.—Cf. also an accident waiting to happen.

The earliest occurrence of the extended phrase that I have found refers to unmarried pregnancy; it is from the account of a case heard at the Court of Common Pleas, published in the Kentish Gazette (Canterbury, Kent, England) of Tuesday 29^th November 1808:

Mr. Sergeant Sheppard, in stating the case, observed, that accidents would happen in the best regulated families; in the present instance, the defendant’s dairy-maid had altered her shape, and Sarah Hughes, the deluded damsal [sic], was removed to the house of a person named Hughes, where the defendant, the father of her child, asked her what doctor she would have. [&c.]

The second-earliest occurrence that I have found is from the Utica Patriot, & Patrol (Utica, New York) of Tuesday 25^th August 1818:

To Correspondents.—[…] The author of the second beautiful acrostic on the name of Hastings on [?] music, and who was notified in our last, that the first was unfortunately torn to pieces, is again informed, that the second has met a similar fate. “Accidents will happen in the best regulated families,” says Poor Richard. ¹

¹ Poor Richard is the name of the supposed author of a series of almanacs published by the statesman, inventor and scientist Benjamin Franklin (1706-1790) between 1732 and 1757, in which advice was dispensed in the form of maxims. Franklin’s maxims were popularised after a number of them were collected in a preface to the almanac for 1758, where each was followed by the phrase as Poor Richard says.

The phrase is used humorously in the following letter, published in The Morning Chronicle (London, England) of Monday 28^th December 1818—puppyism denotes conceit:

PUPPIANA.
“Eye Nature’s walks, shoot Folly as she flies,
And catch the manners living as they rise.”—Pope. ²

To the Editor of the Morning Chronicle.
Sir,
If ever the puppyism and dandyism of the day were likely to be swept from the Metropolis, I think it is now, when all rational parts of the community join in the common cry of ridicule and detestation which the conduct, character and attire, of our modern Beaus are calculated to excite; but permit me to throw out a short suggestion, which may, perhaps, account for the strange metamorphose that has taken place in the rising generation.
One of those accidents (if accident it may be called) recently befel [sic] me, which will sometimes take place even in the best regulated families, viz. the presentation to me of a son and heir. Think, Sir, how the fond feelings of the enraptured, newly created Papa must have been shocked, when a proposition was made by the nurse, to facilitate the infant’s nutrition, by placing at its mother’s breast a Puppy! Yes, Sir, a Puppy—and in recommendation of the measure, it was urged that it was only an old custom newly revived, and was now-a-days quite common. I thought it very common, and therefore forbad it, from the idea that our modern young Gents derive their canine nature from this practice.
Whether this idea is founded in correctness I shall leave any Man-midwife or Midwife’s Man to declare, who shall think fit; and
I have the honour to be your humble servant,
                                                                       J. A. D.

² This is a quotation from An Essay on Man (1733-34), by the English poet Alexander Pope (1688-1744).

A person signing themself ‘Fanny Fanciful’ used mistakes will happen in the best-regulated families in On Fans, a letter to the Editor, published in The Lady’s Magazine; or, Entertaining Companion for the Fair Sex, appropriated solely to their Use and Amusement (London: Printed for G. G. J. and J. Robinson) of November 1791:

In the midst of our hurry and bustle, you know it is a privilege with us ladies always to forget something; for instance, a pair of gloves, a play-book or a prayer-book, a smelling bottle, or a fan. And mistakes will happen in the best regulated families; I have taken my opera fan to church, and have paid a funeral visit with my country dances.