The phrase eureka moment (also moment of eureka) designates a moment of sudden discovery, inspiration or insight.

This phrase occurs, for example, in New music theatre in utero, by Matthew Lorenzon, published in RealTime (Sydney, New South Wales, Australia) of April/May 2011 [No. 102, page 36, column 1]:

Composer Angus Grant found the ideal subject matter for his operetta Contact! in suburban netball—dramatic, quintessentially Australian and just a little daggy. The cast, consisting of a seven-girl netball team, their coach Bev and her son Bevan, induce a few Eureka moments as the traditional opera elements of chorus, recitative and aria are used to add expressive and humorous emphasis to netball colloquialisms and high school vernacular.

In the phrase eureka moment (also moment of eureka), eureka (from Greek εὕρηκα, translating as I have found (it)) is the reputed exclamation of the Greek mathematician and physicist Archimedes (?287-212 B.C.) when he realised, during bathing, that a body immersed in a fluid is buoyed up by a force equal to the weight of the fluid displaced by the body.—Cf., below, quotation 3.

Note 1: The correct transliteration of Greek εὕρηκα is heureka. Greek εὕρηκα is the first person singular perfect of the verb εὑρίσκειν (heurískein), meaning: to find, to discover.—Cf. the English adjective heuristic, meaning: helping to discover or learn.

The earliest occurrences that I have found of the phrase eureka moment (also moment of eureka) are as follows, in chronological order:

1-: From Essay on the Creative Imagination (Chicago: The Open Court Publishing Company, 1906), by the French psychologist Théodule Ribot (1839-1916), translated by Albert H. N. Baron [Chapter 7: The Utopian Imagination; page 302]:

The legend relates that Buddha, possessed with the desire of finding the perfect road of salvation for himself and all other men, gives himself up, at first, to an extravagant asceticism. He perceives the uselessness of this and renounces it. For seven years he meditates, then he beholds the light. He comes into possession of knowledge of the means that give freedom from Karma (the chain of causes and effects), and from the necessity of being born again. Soon he renounces the life of contemplation, and during fifty years of ceaseless wanderings preaches, makes converts, organizes his followers. Whether true or false historically, this tale is psychologically exact. A fixed and besetting idea, trial followed by failure, the decisive moment of Eureka! then the inner revelation manifests itself outwardly, and through the labors of the master and his disciples becomes complete, imposes itself on millions of men.

Note 2: “decisive moment of Eureka!” translates “moment décisif de l’εὕρήκα” in the French original text by Théodule Ribot, Essai sur l’imagination créatrice (Paris: Félix Alcan, 1900) [Chapitre 7: L’imagination utopique; page 252].

2-: From Morale in War and After, by the U.S. psychologist Granville Stanley Hall (1844-1924), published in the Psychological Bulletin (Lancaster (Pennsylvania) and Princeton (New Jersey): Psychological Review Company) of November 1918 [Vol. 15, No. 11, page 367]:

The most universal greeting the world over consists in inquiries about our health and perhaps even that of those nearest to us, as if all must recognize its cardinal importance. Now, real health is not merely keeping out of the doctor’s hands, but its cult aims at keeping each at the very tip-top of his condition so that he feels full of the joy of life (euphoria) and capable of doing or suffering anything. Most of the world’s work is done on a rather low hygienic level, but its great achievements, the culminating work of the leaders of our race, have been the product of exuberant, euphorious, and eureka moments, for a man’s best things come to him when he is in his best state.

3-: From a review of Insight and Outlook. An Inquiry into the Common Foundations of Science, Art and Social Ethics (New York: The Macmillan Company, 1949), by the Hungarian-born British author Arthur Koestler (1905-1983)—review published in The Scotsman (Edinburgh, Midlothian, Scotland) of Thursday 11^th August 1949 [No. 33,137, page 9, column 3]:

Perhaps the most famous bath in recorded history—apart, of course, from those enjoyed by Mr Churchill while dictating urgent dispatches in the conduct of the Unnecessary War—was that of Archimedes in ancient Greece. As the great mathematician was stepping from his bath, all of a sudden his problem was resolved: he saw in a flash that the volume of a solid could be measured by the volume of the liquid displaced by its immersion. Overjoyed, he dashed out and ran through the streets shouting “Eureka, Eureka.” Mr Koestler would say that Archimedes had found a “junctional concept,” so that two independent and mutually exclusive mental fields were “bi-sociated.”
The concept of bisociation (dual association) is, Mr Koesler [sic] says at once, fundamental to his book; it is much later that he calls it the “Eureka Process.” The reader of this brilliant but somewhat dazzling volume will hear much of the Eureka Process and, indeed may have occasion with good fortune and application to follow many quiet Eureka moments with one much-postponed grand Eureka: when he finds just how two phases of the book go together.

4-: From a review of Jean Santeuil (London: Weidenfeld and Nicolson, 1955), by the French novelist, essayist and critic Marcel Proust (1871-1922), translated by the British publisher and translator Gerard Hopkins (1892-1961)—review by the British author Philip Toynbee (1916-1981), published in The Observer (London, England) of Sunday 2^nd October 1955 [No. 8,570, page 11, column 6]:

To those who know Proust already, the fascination of reading Jean Santeuil will lie in the infinite speculations which it must arouse on the nature of the “lift,” the qualitative change which transformed this clumsy work into a masterpiece. What happened in the interval between the abandonment of Santeuil, the beginning of Swann? What divine grace descended on the hitherto un-aureoled head of that young man? […] We can say one thing—that the change from third to first person must have aided Proust in his act of self-liberation. We think of him now as the greatest fictional master of first-person narrative who has ever written, and his discovery of the apparently limiting, the actually liberating “I” must have been a eureka moment in his life.