Of British-English origin, the colloquial phrase a blind bit of —— is used in negative constructions with a following noun to mean a single ——, any ——. The nouns most commonly used in those constructions are notice and difference.

This phrase occurs, for example, in Grealish’s treatment of RTE’s Jim Fahy said the game was up, by Andrew Lynch, published in the Evening Herald (Dublin, County Dublin, Ireland) of Tuesday 16^th September 2008 [Vol. 116, No. 222, page 12, column 5]—the following is about the Irish politician Ciarán Cannon (born 1965), who was the Leader of the Progressive Democrats (PDs) for a few months in 2008; Trinny Woodall (born 1964) and Susannah Constantine (born 1962) are a duo of British fashion advisors:

During his five months as leader, the mild-mannered Galway man has failed to capture the public’s imagination.
After his colleagues berated him for not being visible enough, Cannon spent the summer firing out statements on everything from the Lisbon Treaty to the recession. Nobody paid a blind bit of attention because everybody knew that the PDs’ view is about as relevant as Trinny and Susannah’s.

The earliest occurrences of the phrase a blind bit of —— that I have found are as follows, in chronological order:

1-: From Quest Sinister (London: Grant Richards Ltd., 1922), by the British author and journalist Stuart Petre Brodie Mais (1885-1975) [page 241]:

“When they’re tired of nagging at each other they have a go at me. It does them both good, and I never take a blind bit of notice. Why should I? They keep me for that special purpose. They’re both damned fools, but I happen to like damned fools. It’s like watching a dog-fight. So long as I don’t interfere too much, it’s quite fun.”

2-: From Hyde Park Orator (New York: G. P. Putnam’s Sons, [1934]), by the Irish political orator Bonar Thompson (1888-1963) [page 181]:

“There is only one thing for it,” I said to Smith. “I must cast a few pearls among swine. Let us go into this public-house and I will recite something.”
I approached the landlady. “Have you any objection, madam,” I asked, “to my giving a recitation to your guests and passing the hat round at the end?” “Do so, if you like,” replied the good lady, “but I don’t think you’ll do much good.” “One can but try,” I remarked. Addressing the whole room collectively, I announced my intention of rendering a dramatic poem entitled “Fra Giacomo” by Robert Buchanan.
I stood upon a little platform and let it go. Hunger and desperation helped me to impart the proper wildness and dramatic frenzy into the character of the unfortunate victim of woman’s frailty. In vain. The sturdy artisans sat at their tables in supreme unconcern. Not a muscle of their faces moved. No one, to use a proletarian phrase, took a “blind” bit of notice. The hat went round, and returned—empty.

3-: From The Battle of the Flames, in Front Line 1940-41: The Official Story of the Civil Defence of Britain (London: His Majesty’s Stationery Office, 1942) [page 26]—the following is from a narrative from an auxiliary fireman; W.A.F.S. stands for Women’s Auxiliary Fire Service:

“On 7^th September we took our pumps to East India Dock, to Rum Wharf. The first line of warehouses was ablaze from end to end… I walked down between the two warehouses by myself. Half-way down was a staff car in the middle of the causeway. Standing nonchalantly by it was a young W.A.F.S., outwardly not taking a blind bit of notice of the stuff that was falling pretty thick all round. Seeing her I strolled past as if I was used to walking out in the middle of falling bombs every Saturday afternoon.”

4-: From None but the Lonely Heart (New York: The Macmillan Company, 1943), by the British novelist Richard Llewellyn (1906-1983):

[page 70]: He pulled the playing lever, very gentle and fancy, as if he had the whole secret of it off by heart, standing back with his hands on the woodwork, like most of them done, pressing down till his shoulders come up nigh as high as his head, watching that there ball like as if it had all the hopes of the world sliding round on it. Frowning, he was, screwing his face up, shoving one shoulder forward when it looked as if the ball was going to miss hitting one of them little numbers, wagging his hips about when a half inch miss come up, as though he expected to give his play that bit extra by chucking his self about, not that the ball ever took a blind bit of notice of him nor nobody else.
[page 353]: Taz looked through the window again, and the car started dodging traffic. He was holding the gun in one hand and the bullet in the other, like a proper Joe Soap, but nobody was taking a blind bit of notice of Him.

5-: From It Was Bloomin’ Magic!, published in the Sunday Pictorial (London, England) of Sunday 12^th December 1943 [No. 1,500, page 12, column 2]:

THIS balloon, they flies this balloon from a field right next door to our billets and for the past two years it goes up and down, up and down, and nobody in our mob takes a blind bit of notice.

6-: From the Norwood News (London, England) of Friday 25^th May 1945 [No. 4,141, page 3, column 2]:

MEETING HIS OLD FRIENDS
The Rev. L. Vick at Anerley

The Rev. Laurance Vick, formerly curate at Holy Trinity Church, Anerley, who left some time ago to serve as a chaplain in H.M. Forces, and whose recent marriage was reported in the “News,” brought his bride to meet his old friends at Anerley on Sunday, and preached at the morning and evening services of his former church.
[…]
“The reasons for not coming to church that were given in padre’s hour * are varied,” he said. “Some are tragic in their misconception—‘Come to church and the parson pinches the collection!’ ‘I don’t go to church because the parson doesn’t come to see me,’ ‘Go to church, and nobody takes a blind bit of notice of you, but go to the Methodists, and somebody at once comes up and says “Hallo” and shakes hands with you.’”

* The expression padre’s hour designates a weekly hour of religious instruction provided by chaplains to British-Army units.