A loan translation from the German noun Ohrwurm, earworm denotes a catchy song or melody that keeps repeating in one’s mind, especially to the point of irritation.

ORIGINAL MEANING OF EARWORM: AN EARWIG

The noun earworm originally denoted an earwig. This noun is first recorded in Of all the kinde of beastes, Cattell, and foules in India, the 45^th chapter of Iohn Huighen van Linschoten. his Discours of Voyages into y^e Easte & West Indies (Printed at London: By [John Windet for] Iohn Wolfe printer to y^e Honorable Cittie of London, [1598]):
—Here, earworm may be a loan translation from Dutch oorwurm (attested in 1351), as this book is a translation by William Phillip (fl. 1596-1619) of Itinerario: Voyage ofte schipvaert van Jan Huygen van Linschoten naer Oost ofte Portugaels Indien, 1579-1592, by the Dutch merchant and historian Jan Huygen van Linschoten (1563-1611):

Moathes & wormes which créepe and eate through mens cloathes, are there in great aboundance, whereby men must vse no more cloathes nor linnen in those countries then that he necessarily and dayly weareth on his back, otherwise they are presently moath-eaten and spoyled. They can hardly kepe any paper or bokes from wormes, which are like eare wormes, but they do often spoyle & consume many papers & euidences of great importance.

Interestingly, earworm occurs as a translation of the German noun Ohrwurm, in the sense of an earwig, in the following passage from Flashlights on Nature (New York: Doubleday & McClure Co., 1898), by the Canadian science writer and novelist Grant Allen (1848-1899):

I will gird on my sword to do battle for the right, and rush in, a scientific St. George, in defence of the innocent but persecuted earwig.
That my hero (or heroine) has a bad name in the world I am not careful to deny. Calumny has dogged it from its earliest days. Its very name enshrines a myth which is in itself a libel. It is called earwig, gossips will tell you, because it creeps into the ears of incautious sleepers in the open air, and so worms its way to the brain, where, if you will believe the purveyors of folk-lore natural history, it grows to a gigantic size, “as big as a goose’s egg,” and finally kills its unhappy victim. It is true, science knows nothing of this form of brain-disease; it has tried the case before an impartial tribunal, and the earwig has left the court without a stain on its character. Some etymologists have even endeavoured to persuade us that the name earwig itself is but a corruption of ear-wing, a word which they suppose to be derived from the shape of its flying organs. There, however, our philologists are surely crediting the people with more knowledge than they possess; very few gardeners or countrymen are aware that earwigs have wings, while the general public never sees them flying. Besides, the German name Ohrwurm, or “ear-worm,” and the French Perce-oreille, or “pierce-ear,” suffice to show that the myth is not confined to our own country. All over the world this harmless and on the whole beneficent creature (for he is a good scavenger) is regarded with superstitious fear and aversion; all over the world he is ruthlessly destroyed whenever found; and modern science alone is the first to attempt the herculean task of rehabilitating him.

EARLY OCCURRENCES OF EARWORM IN THE CURRENT ACCEPTATION

These are the earliest occurrences that I have found of earworm used in the sense of a song, tune or (in early use) words that persistently stay in a person’s mind:

1-: From Human Linguistics, by the U.S. linguist John Robert ‘Haj’ Ross (born 1938), published in Contemporary Perceptions of Language: Interdisciplinary Dimensions (Washington, D.C.: Georgetown University Press, 1982)—Haj Ross applied earworm to the words of certain poems:
—In a headnote, the Editor, Heidi Byrnes, explained: “The typographical format requested, designed, and employed by the author to make his personal statement is intended by him to reflect the style of an oral paper.”:

     i’ll tell you how i came to have these disrespectful
thoughts    i’ve been studying poetry for four years    and
here’s the way the enterprise proceeds for me    i find some
   poem which    for who knows what reason    comes into my
blood    it just knocks me over    and i want to find out why
     because essentially that’s magic    a poet can do magic
a poet does something to the words which you and i use in our
   everyday interactions    and puts them there and they become
what they call in german an ‘ohrwurm’    an ear worm    you
     can’t get it out of your head

2-: From Strange Memes: Language Viruses, the 11^th chapter of They Have a Word for It: A Lighthearted Lexicon of Untranslatable Words and Phrases (Los Angeles: Jeremy P. Tarcher, Inc., 1988), by the U.S. author Howard Rheingold (born 1947):

Ohrwurm (German): A tune or melody that infects a population rapidly. [noun]
If a meme is a cluster of semantic symbols that propagates through a human population in a social manner, similar to the way a gene is a combination of biochemical symbols that propagates through a human population in a genetic manner, a sudden, wildly popular new addition to “the top 40” can be seen as a kind of meme. When the medium of radio and the recording industry that grew up alongside it created a system for propagating musical themes through a population, a new phenomenon became possible—the “overnight hit.” The idea of a hit isn’t untranslatable, since most cultures have a word for the winner of a competition. But the idea of a tune, a melody, a combination of musical sounds that seems to be on everybody’s lips at the same time, that spreads through a society as rapidly as a respiratory infection and that seems to invasively seize and occupy space in people’s minds until they finally succeed in forgetting it merits a word of its own.
The Germans use the word Ohrwurm (rhymes with “door worm,” where the “w” is pronounced like a “v”) to denote these cognitively infectious musical agents. Whenever somebody complains to you that she just can’t keep the latest pop tune from running through her head, tell her she can dispel it by calling it by name and by thinking about the original German meaning, which captures some of the mnemonically parasitical connotations of the word, for Ohrwurm literally means “ear worm” and is also used to refer to a kind of worm that can crawl into the ear.