Mr O’Madden Burke winces at the answer to Lenehan’s riddle about the Rose of Castile. The earliest examples of his expression "a strong weakness" date from the 17th century, but in religious contexts:1
There is a strong weakenesse, and there is a weakeness that is weake in deed.
Thomas Tuke, Picture of a True Protestant (1609), p. 80
Therefore it is a very strong weakness or wilfulness in some who love to turn Straws into Trees and Feathers into Birds, and not to leave things as Christ hath left them …
Samuel Annesley, Casuistical Morning-Exercises (1690), Sermon 10, p. 242
By the 19th century it had developed (especially in Ireland) into yet another well-worn catchphrase. It served as an expression describing cowardice, then love:
Belpêche (the strong weakness of whose timidity was overcome by the more powerful curiosity of seeing the result of his rival's adventure)
Thomas Colley Grattan, High-ways and by-ways (1827), p. 233
It is the bane of youth — the heritage
Breeding us cowards in our own despite,
Unmindful of the war we have to wage;
Yea, this strong weakness doth o'ercome us quite,
Making us bear with wrong, and pain, and slight …
Charles Whitehead, The solitary (1831), p. 49
It was, indeed, Mrs. Candy; won to the imprudence by the strong weakness of love, she had prompted her maid to touch upon the future fate of her mistress, herself hid the while among the bushes.
Douglas William Jerrold , Men of character (1838), p. 138
But soon it was (often humorously) used to allude to the strong national weakness of drink.
He came to Ireland on the strength of his equity law, but he got a constitutional weakness (laughter), and he did not go away on the strength of his constitutional law (laughter), unless it could be designated in the language of the west2 a strong weakness (great laughter).
Freeman's Journal (1843), 29 June
[T]he coachman, an old rascal [...] who had what the Irish call a 'strong weakness' for liquor of any description.
Dublin University Magazine (1862), December p. 714/2
He was constantly overtaken by what he termed "a strong wakeness". When seized by one of these "turns", as he called them, a seat on the nearest stone, and a long pull and a strong pull at Maurice's flask were the only remedies ...
Bithia Mary Croker, Pretty Miss Neville (1883), p. 75
The paradox easily survived into the 20th century:
A Feminist aphorism
"We, of the weaker sex, are stronger than the stronger sex, because of the strong weakness of the stronger for the weaker sex."
Idaho (Boise) Daily Statesman (1914), 29 January, p. 4
Beckett provides its apotheosis:
One day, in a positive geyser of confidence, he gave me an account of one of these "moving pauses". He had a strong weakness for oxymoron. In the same way he over-indulged in gin and tonic-water.
Samuel Beckett, More Pricks than Kicks (1934; ed. 1972), p. 38
Curiously, O’Madden Burke almost strips the expression of its oxymoronic character by using it in a purely medical sense where a strong weakness usually leads to fainting3:
"My child! my poor child!" exclaimed Mrs. O'Flaherty, sinking into a chair and fainting. "Oh, botheration, isn't it enough to dhrive any one on earth slap out of their senses, if they had fourteen instead of seven," said Judy, bursting into a passion of tears. "The daughter gone off, God alone knows where; and the mother, God be good to the poor demented crater, fainting away as dead as a herring! What on earth will I do to get her out of this fit?" And Judy ran away to get feathers to burn under the nose of her mistress, which having done, she rubbed her hands and sprinkled her face with water. "Murder! murder!" cried Judy, the tears running down her face, "did any one ever see such a strong weakness? Ma'am, Ma'am, come to yourself a bit! just open your eyes.
Marguerite Countess of Blessington, Country Quarters (1850), p. 286