How long has this phrase been around?
I feel a bit cheesy using it in an essay because it reminds me of the film.
English Language & Usage Stack Exchange is a question and answer site for linguists, etymologists, and serious English language enthusiasts. It only takes a minute to sign up.Sign up to join this community
The phrase as it is commonly known today was coined by Winston Churchill on 18 June 1940 according to a note in Oxford English Dictionary.
The OED defines this as a phrase under the headword fine.
P6. one's finest hour: the time of one's greatest, most noble, or most admirable achievement. Now often in negative contexts, as not one's finest hour.
So bear ourselves that if the British Commonwealth and Empire lasts for a thousand years men will still say, ‘This was their finest hour.’
To complement RaceYouAnytime's answer, a Google ngram search indeed indicates that the popularity of the phrase "finest hour" jumps abruptly from almost nothing to about half its modern level in 1940, presumably as a result of Churchill's speech:
Looking at the few scattered hits from 1939 and earlier, most of them appear to be either misdated modern sources (e.g. this text is clearly not from 1881, since it explicitly refers to the events of 1940), new additions to old texts (e.g. this modern foreword to an old book) or, in several cases, OCR errors for "finest flour" (e.g. here). That said, I did find a few apparently genuine uses of the phrase from before 1940, such as:
What is the finest hour of the day to you? Hardly noon or midnight. I think you would answer, “Twilight”—the holy hour when day and night meet in solemn, ...
The Things Not Seen, Richard Tillman Vann, Sunday School Board of the Southern Baptist Convention, 1931
Thus does he forfeit and never meet the finest hour of all the twenty-four — that hour in which the birds flutter out from their ...
"Old Limber": Or, The Tale of the Taylors, DeLong Rice, McQuiddy Printing Company, 1921
The finest day that I have ever spent — and I have only spent three or four days in Calgary — but the finest hour that I have spent in Calgary was the hour that I experienced in judging the fat lambs for about thirty boys and girls in your judging arena at the Calgary Winter Fair in November.
THE HUDSON HIGHLANDS “AT THE GLOAMING," THE FINEST HOUR FOR MOUNTAIN AND RIVER SCENERY
The Official Guide of the Railways and Steam Navigation Lines of the United States, Porto Rico, Canada, Mexico and Cuba, National Railway Publication Company, 1910
Note that all of these early uses appear to be quite literal, describing a specific time of day as the most picturesque or pleasant, or a specific time in the past as the most memorable. It does seem likely that the now common metaphorical use of the phrase, to describe one's crowning moment of achievement, was coined or at least introduced into common use by Churchill.
From Poor Isabel in the March 1865 Harper's:
The young woman, decently but poorly clothed in black, was something almost at variance with the gladness and sparkle of that finest hour of June; the budding briers, the opening, honeysuckles, the twittering birds, all seemed to belong to a different world from that in which she had lived.
Also, in the 21 June 1866 The Round Table:
It is melancholy to see men who need all the exercise and air that they can get, as well as all the rest for their eyes, and all the modifying and exhilarating influences of companionship, riding down town in a crowded car or omnibus, in the finest hour of the morning; sitting or standing in glum silence, and endeavoring to read a newspaper, the small letter and bad printing of which, joined to the motion of the vehicle, form the most effective possible union of means for injuring the sight.
In print and in English, the phrase 'finest hour(s)' has been in use for at least 200 years. The first use I found, from 1817, in Volume 1 of The Hero; or, the Adventures of a Night: A Romance ("Translated from the Iroquese; from the Iroquese into Hottentot; from the Hottentot into French; and from the French into English"), is figurative in scope although superficially literal in that it refers to "midnight":
"Oh! yes, yes!" said the monk, interrupting him, "I know as well as you do, that in an English romance midnight is the finest hour of the day! But," ....
The next use, from 1823, is a mundane literal use in a didactic moral text, An Inquiry, with a View to Ascertain how far Nature and Education Respectively Determine the Moral and Intellectual Character of Man:
During four months of the winter, from eleven to two, being commonly the three finest hours of the day, occurring betwixt the morning and evening damps or fogs, ....
Next comes another figurative use, in a book review by Edgar Allan Poe from Graham's Magazine, 1841. This is a specific application of the general figurative sense given by OED's lexical definition of 'finest hour' ("the time of one's greatest, most noble, or most admirable achievement"):
Upon the whole, it is fully as good a poem as Pope could have written, upon the same subject, in his finest hour of inspiration.
A semi-literal use occurs in the 1886 volume of The Nineteenth Century, in an article by Mrs. Arthur Kennard titled "Gustave Flaubert and George Sand":
There is no doubt, when we grow old and reach the sunset of life (the finest hour for tones and harmonies of colour), we form new ideas of everything, and above all of affection.
The next instance I found, from the 1897 publication of Wayside Courtships, by Hamlin Garland, seems to communicate a generalized literal, rather than figurative, sense:
He talked on slowly. He asked her what her plans were.
"To teach and to live," she said. Her enthusiasm for the work seemed entirely gone.
Once he said, "This is the finest hour of my life."
In the 1902 Paul Kelver, a novel by Jerome K. Jerome, another use with the figurative meaning defined in OED appears:
"You would be a fool if you did," he went on. "One's first success, one's first victory! It is the lover's first kiss." ....
"Ah yes; one's first success, Paul! Laugh, my boy, cry! Shut yourself up in your room, shout, dance! Throw your hat into the air and cry hurrah! Make the most of it, Paul. Hug it to your heart, think of it, dream of it. This is the finest hour of your life, my boy. There will never come another like it...."
The 1911 semi-autobiographical Anglo-American Memories features another use with the figurative sense given by OED:
It [a speech against slavery] was such a triumph as comes to a man once in his career, and once only — the finest hour in Phillip's life. He never reached a greater height of oratory, nor an equal height of devotion. For his triumph was over himself.
In the 1920 People of Destiny; Americans as I Saw Them at Home and Abroad, the figurative sense appears again:
On that first night of mine [lecturing at Carnegie Hall] I would have sold myself, with white shirt, cuff-links, and quaking body, for a two-cent piece, if any one had been fool enough to buy me and let me off that awful ordeal. And yet, looking back on it now, I know that it was the finest hour of my life, and a wonderful reward for small service, when all those people rose to greet me, and there came up to me out of that audience a spiritual friendship so warm and generous that I felt it like the touch of kindly hands about me, and recovered from my fright.
The sprinkling of uses of the phrase 'finest hour' in both figurative and literal senses which I have documented between 1817 and 1920, as well as others I have no doubt overlooked, led to later uses, such as the 1940 use in Churchill's famous speech (as noted in another answer), as well as use as the title of the 2016 disaster film The Finest Hours.
However, in the microcosm of the use of the phrase, which may also include use in your essay ("I feel a bit cheesy using ['finest hour'] in an essay because it reminds me of the film" — OP), Churchill's 1940 admonition applies:
Of this I am quite sure, that if we open a quarrel between the past and the present, we shall find that we have lost the future.