Of nautical origin, the term Mother Carey’s chicken designates the storm petrel. It is first attested in An Account of a Voyage round the World, in the years MDCCLXVI, MDCCLXVII, MDCCLXVIII, and MDCCLXIX. By Philip Carteret, Esquire, Commander of his Majesty’s Sloop the Swallow, as edited by John Hawkesworth (circa 1715-1773) and published in London in 1773:

1767, April. Saturday 18: From the time of our clearing the Streight [of Magellan], and during our passage along this coast, we saw a great number of sea birds, particularly albatrosses, gannets, sheerwaters [= shearwaters], and a thick lumpish bird, about as big as a large pigeon, which the sailors call a Cape of Good Hope hen: they are of a dark brown or blackish colour, and are therefore sometimes called the black gull: we saw also a great many pintado birds, of nearly the same size, which are prettily spotted with black and white, and constantly on the wing, though they frequently appear as if they were walking upon the water, like the peterels, to which sailors have given the name of Mother Carey’s chickens; and we saw also many of these.

The name Mother Carey is probably a mistranslation of post-classical Latin Mater Cara or Italian Madre Cara (attested in 1364), meaning dear mother and used as epithets for the Virgin Mary. The storm petrel was thought by sailors to be a harbinger of bad weather sent by the Virgin Mary.

The following explanations are from American Notes and Queries (Philadelphia, Pennsylvania) of 1^st June 1889:

WHENCE THE NAME “MOTHER CAREY’S CHICKENS”?

The names “Mother Carey’s Chickens” and “Stormy Petrel” are applied by sailors to the bird known to ornithologists as the Thalasidroma pelagica. Despite its name, it is a tiny creature, scarcely larger than a lark, the very smallest web-footed bird known; of a sooty-black color, with a little white on its wings and tail, and so thoroughly given up to an Esquimau diet of fish and whale blubber as to be extremely unpleasant to approach.
[…]
The name may be regarded as an English corruption of “Mater Cara” (Dear Mother), the appellation bestowed by Italians upon the Virgin, who from time immemorial has been regarded as the special patroness of mariners.
The halcyon, which has been in a measure identified with the petrel, is familiarly known on the Mediterranean coast by the French as “l’oiseau de Notre Dame¹,” “avis Sanctae Maria²,” and by the Sardinians as “ucello pescatora Santa Maria³.” Birds of this class, which are thought to give friendly warning at sea of approaching storms, are naturally regarded as birds or messengers of the Virgin, who, in the character of their patroness, has the safety of the mariner in her special keeping.
This is the more probable, when we bear in mind the great power over the sea attributed by the Romish Church to the Holy Mother, who is the sailor’s “Stella Maris⁴,” whose protection he invokes in song when danger threatens on the deep […].
The modern sailor is, as every one knows, full of the oddest superstitions, which have survived among seafaring men from the earliest ages of navigation. And Jack Tar of to-day pays his respects to auguries in the same manner as did the Greek sailor of Aristophanes’ time, more than two thousand years ago. Peisthetairus, in “The Birds,” says: “Some one of the birds shall always foretell to him that consults them about the voyage; ‘now sail not, there will be a tempest; now sail, gain will ensue.’” We have read that Alexander was led on to a victory over his great adversary, Darius, by the encouraging flight of an eagle, and that Romulus “builded his kingdom by flying of fowls and soothsaying.”
[…]
As these small birds are known as “Mother Carey’s Chickens,” it is but a natural consequence that the great black petrel, belonging to the same family, frequent in the Pacific Ocean, and a ravenous feeder upon dead whales, should be called “Mother Carey’s Goose;” and when it snows the sailors say “Mother Carey is plucking her goose,” supposed to be a facetious interpretation of the old German legend that described the snow as the feathers falling from the bed of the benignant goddess Hulda, when she shook it up in making it.
The name “Mother Carey’s Chickens” came to be also applied to the mobs which thronged the streets of Paris during the first great French Revolution; they were so called because their appearance, like that of the “stormy petrel,” was the foreboding of woe, the heralding of a tumult and political stormy weather. Carlyle, in his chapter on the “Insurrection of Women,” pictures one of these crowds which issued from the guard house “Like snow-break from the mountains, every staircase a melted brook; it storms tumultuous and wild-shrilling; in the rear stones already fly; women, copiously escorted by hunger and rascality, press on, while guidance there is none but two drumsticks—a slow-moving chaos, the modern saturnalia of the ancients.”

¹ l’oiseau de Notre Dame: Our Lady’s bird
² Latin avis Sanctae Maria: Saint Mary’s bird
³ Italian ucello, bird, pescatora, fisher
⁴ Latin Stella Maris: the star of the sea