In eligible bachelor, the adjective eligible means suitable as a partner in marriage.
—Cf. the answer to a maiden’s prayer.

These are, in chronological order, the earliest occurrences of eligible bachelor that I have found:

1-: From Horatio in Search of a Wife. A Tale of Modern Times (Leeds: Published by John Saunders, 1830), by Anna Maria Morgan and Hannah Maria Jones:

Matilda, indeed, was fast verging towards that age when the enviable appellation, “maid,” is disfigured by ill-natured people persisting to place the epithet, “old,” before it, and she could not but feel, bitterly feel, that the chance of her retaining this hateful title for the remainder of her life was increased tenfold, now that it was publicly known that her father could not give her a fortune.
It was in vain that Letitia, who seldom did a good-natured action, but who, on this point, entered fully into the feelings of her mother and sister—it was in vain that she invited her to all her parties when there were any eligible bachelors expected—in vain she echoed all Mrs. Lane’s puffs, either direct or indirect, on her sister’s accomplishments—in vain she whispered to all her male acquaintance, that Matty, if she could be prevailed upon to think of matrimony, would make the very best of wives in the world.

2-: From The Maids, Wives, and Widows’ Penny Magazine, and Gazette of Fashion (London, England) of Saturday 19^th January 1833:

NOCTES TWANKAYANÆ.
No. V.

Board of Green Tea—Present, Miss Bluemantle, Mrs. Bloomer, Mrs. Modish, and Miss Harmonica.
Miss Bluemantle. Doubtless most of our “Wives” are on tiptoe to hear Bloomer’s opinion of Lady Dacre’s Recollections of a Chaperon.
Mrs. Bloomer. Nay, I fear it will be forestalled by many of them, as the book has most likely by this time passed through the hands of almost every Mamma with grown up daughters, in England. But, I fear, they will be sadly disappointed; instead of being instructed in all the wily arts of catching eligible bachelors, they will find a few tales “founded on facts”—clever, very clever, but not the description of matter that might have been expected from the title page.

3-: From Horatio Sparkins, published in Watkins Tottle, and other sketches, illustrative of every-day life and every-day people (Philadelphia: Carey, Lea & Blanchard, 1836), by ‘Boz’, i.e., the English novelist Charles Dickens (1812-1870):

Miss Teresa Malderton was a very little girl, rather fat, with vermilion cheeks: but good-humoured, and still disengaged, although, to do her justice, the misfortune arose from no lack of perseverance on her part. In vain had she flirted for ten years; in vain had Mr. and Mrs. Malderton assiduously kept up an extensive acquaintance among the young eligible bachelors of Camberwell, and even of Wandsworth; and Brixton on Sunday, to say nothing of those who “dropped in” from town. Miss Malderton was as well known as the lion on the top of Northumberland House, and had about as much chance of “going off.”

4-: From Sketches of the Higher Classes of Colored Society in Philadelphia (Philadelphia: Merrihew and Thompson, 1841), by Joseph Willson:

On occasions of appointed entertainments, which are usually in the form of evening parties, the greatest order and neatness of management is observed. There is always a first and second table, both appropriately and well stored, and in a manner generally unexceptionable. The guests at the parties consist only of such individuals or families as are accustomed to entertain their host or hostess, for the time, in a similar manner in their turn. Those who are all things in reception and nothing in return,—strangers, agreeable, entertaining and eligible bachelors, and sojourners in the city at board, always of course excepted,—are considered detrimental to the harmony of social intercourse; and, consequently, are very apt to be neglected in the list of invitations, that may be issued! In this way they manage to collect a very agreeable company, all perhaps on terms of perfect agreement and intimacy.

5-: From John O’Groat Journal (Wick, Caithness, Scotland) of Friday 11^th November 1842:

THE NATURAL HISTORY OF COURTSHIP.
CHAPTER XIII.—ON COURTING BY ADVERTISEMENT.

“Tea or coffee this morning?” inquires the solicitous Priscilla.
“Neither, just now; as I am reading the Morning Post I imbibe the necessary inspiration from milk and water. You can order strong coffee with the Times.”
The elderly gentlewoman took up her own peculiar print, the Herald. While placidly perusing its pages, her equanimity was suddenly disturbed; that calm countenance slowly wound itself up into an expression of contempt; that mild grey eye flashed forth a beam of indignation; that rigid frame was agitated like unto the seething and boiling of the urn upon the table. In my alarm at so unusual an ebullition, I inquired what was the matter, and looked for the Preston salts.
“I blush for my sex!” exclaimed the disturbed damsel. “Miss Robinson really possesses no principle of honour—Let me read you this paragraph:
“It is whispered in the fashionable circles that a certain young Viscount, lately returned from his travels, is about to lead to the hymeneal altar the lovely and accomplished Miss R. The match, it is rumoured, will be purely one of affection; for the lady was on the eve of marriage to a certain rich commoner, well known in the musical circles; the fascinating financée [sic] having herself broken off the negotiation to favour the suit of the handsome Viscount L——.”
“Pray do not allow a mere matrimonial advertisement (for it is nothing more) to ruffle you,” I said soothingly; “the truth is, Lord Lispen would be a better match for Miss Robinson than Pleinpurse—and you must not be hard upon the coquette, and Mrs Couple, the chaperon, for exercising their vocation by publishing the paragraph.”
“They publish it!”
“Even so; its intention is to apprise Lispen that Rose’s hand is, if he pleases, within his grasp. The whole thing will be contradicted to-morrow by the parties themselves—but what of that? the end will have been answered: Lispen’s connubial pulse will have been felt, his symptoms developed.”
“But are all the similar announcements one daily sees, mere matrimonial advertisements?”
“Nearly so. They are called, in political parlance, ‘feelers’—they suggest to the parties intended to be trapped, what perhaps would otherwise have never entered their brains; and thus it is a common occurrence for the torch of Hymen to be lighted by newspaper paragraphs.”
Not only these, but other announcements of a less suspicious appearance, are rank advertisements for connubial bliss. When you see it stated that at Lady So-and-so’s forthcoming ball, the beautiful Miss —— is to come out, the information is addressed to all eligible bachelors—just as the possessor of a valuable picture invites the inspection of connoisseurs, that whoever can afford the price may take it off his hands.”

6-: From The Cabinet, published in The New Mirror (New York, USA) of Saturday 24^th August 1844:

Brig. Just feel in my pocket for a manuscript! Is it there?
Com. Yes—a letter. What is it about?
Brig. From a man who wants eligible bachelors and old maids brought comfortably together.

7-: From A Year and a Day in the East; Or, Wanderings over Land and Sea (London: Longman, Brown, Green, and Longmans, 1846), by Mrs. Eliot Montauban:

After passing seven days at Suez we went on board the Bombay mail steamer. […]
[…]
Three was a magic number on board the steam prison. A triumvirate of brides were daring the perils of the deep with their favoured Benedicts, edifying the passengers in general with a daily exhibition of the art of cooing and wooing in all its branches and various moods and degrees.
Three fair daughters of Eve appeared in that suffering condition which our immortal bard, in the bliss of ignorance, terms “the pleasing punishment that women bear.”
Three happy wives were en route to their expectant lords, and twice three happier still in the presence of their devoted sposo’s; and three fair lassies, rich in the bloom of girlhood’s beauty, sheltered under the protecting wings of elderly chaperones, were bound for the great matrimonial mart of the East.
Three eligible bachelors, full of buoyant hope and youthful expectation, were journeying to the El-dorado of their separate dreams; and three, in the evening hour of life, forsaking the joys of wedded and filial love, to wander afar in search of gold.
[…]
Rome, the unique, the queen of cities, we reached on a beautiful Spring morning. Swarms of our countrymen and women had preceded us, and we had great difficulty in finding a temporary home. At length we were well accomodated [sic] in a pension bourgeoise, in the Corso, where we found a large society of English, Irish, Scotch, and Americans.
Among the gentlemen, was one eligible bachelor, resembling the caricatures of love among the roses, smiling complacently over a marvellous circumference of flesh, with a pleasing placidity of visage and graceful dignity of demeanour. And many sweet flowers of spinsterhood were there assembled, and interesting specimens of lights from the New World.