The informal British-English expression Hooray Henry designates a lively but ineffectual young upper-class man.
—Feminine equivalent: Sloane Ranger.

The expression Hooray Henry seems to have been coined in the 1950s by the British expert on trad jazz James Godbolt (1922-2013), when he was an agent managing British trad-jazz musicians [cf., below, quotation 4]. And it seems that Godbolt coined the expression Hooray Henry after the earlier Hoorah Henry, coined in 1936 by the U.S. author Alfred Damon Runyon [cf., below, quotations 1 & 4].

The earliest occurrences of the expressions Hoorah Henry and Hooray Henry that I have found are as follows, in chronological order:

1-: From Tight Shoes, set in New York City, by the U.S. journalist and short-story writer Alfred Damon Runyon (1880-1946), published in the Evening Standard (London, England) of Friday 22^nd October 1937—this story was first published in Collier’s: The National Weekly (Springfield, Ohio, USA) of Saturday 18^th April 1936:

It is now along about eight bells in the evening, and not many cash customers are in the Bridle Grill, when all of a sudden in comes a tall, good-looking young fellow in evening clothes, including a high hat and an opera cape, lined with white silk, and who is this fine young but Calvin Colby, who is known far and wide as a great pain in the neck to his loving parents.
He is also known as a character who likes to get around and, in fact, Calvin Colby’s only occupation is getting around. His people are as rich as mud, and to tell the truth, richer, and what is more they are in the Social Register and, in fact, Calvin Colby is in the Social Register himself until the publishers come upon his name one day and see that it is a typographical error.
He is often in the newspapers, because it is really remarkable how Calvin Colby’s automobiles can spill dolls up against telegraph poles along the Boston Post Road, when he happens to strike these obstacles, and the dolls are always suing Calvin Colby for breaking their legs, or spoiling their complexions. It finally gets so there is talk of taking Colvin’s [sic] driving licence away from him before he shatters all the telegraph poles along the Boston Post Road.
He is without doubt strictly a Hoorah Henry, and he is generally figured as nothing but a lob as far as ever doing anything useful in this world is concerned, although everybody admits that he has a nice disposition, and is as good a right guard as ever comes out of Yale.

2-: From Absolute Beginners (London: MacGibbon & Kee, 1959), by the British novelist and essayist Colin MacInnes (1914-1976)—this novel is set in London:

I hurried on, and came on the outskirts of another crowd, and found they were gathered round the Santa Lucia club, which is a BWI * clip-joint about as glamorous as an all-night urinal. There were several hundred milling there: and what added joy to the whole scene, was the presence of newsreel and TV cameras, with arc lamps and the odd flare and flashlight, as if the crowd were extras on a movie lot. […] And as for the Teds and hooligans, well, they can smell a camera, even a Press one, from a mile away, and there’s nothing they like better than seeing their moronic faces next morning in the tabloids, so this was their big opportunity as well.
‘Child!’ shouted someone, and I looked across, and there, standing up in the back of a cream vintage Bentley, was the ex-Deb-of-Last-Year. I struggled across, and found she was with a bunch of Hooray Henries, who seemed, I will say this for them, a bit doubtful if all this was really so dam amusing. And as for the ex-Deb, she leaned out of her vehicle, and said, ‘That crowd’s nothing but a lot of bloody scum.’
‘You’re telling me,’ I said.
‘And what is that place?’ she asked, waving a hand at the Santa Lucia club.
‘A local nitery. You like to take a spin around in there?’ I asked – a bit sarcastically, I must admit, because if the yelling crowd outside didn’t do you, the Spades down in there, if there were any, most certainly would, if you attempted to get in.
‘Certainly!’ she cried, and she spoke up a bit loud even for my liking. ‘I’d love to have a dance with someone African! They’re the best dancers in the world!’
So she told me to get in, which I did thinking, ‘Oh, well!’, and the Henry at the wheel got the car up near the entrance, with everyone, when they saw the ex-Deb and the Henries, imagining, I suppose, that this was some item in the television programme. The ex-Deb and I got out, with a Hooray or two in tow, and shouldered down some steps into the basement area.

[* BWI is the abbreviation of British West Indies.]

3-: From In spite of illness the ‘Doctor’ survived, by ‘P. E.’, published in the Worthing Herald (Worthing, Sussex, England) of Friday 13^th March 1959:

THE West Susses Players were out of luck with their mixture of Richard Gordon’s “prescription for laughter” Doctor in the House.
[…]
Tony Canneaux is a natural. His performance as student medico Simon Sparrow was as easy as they come. But John Erskine was not quite so happy in the part of the life-and-soul-of-the-party student Tony Grimsdyke. He tried hard to assume the devil-may-care attitude of all true Hooray Henrys, but I wasn’t convinced.

4-: From Who are The Ravers?, an article about the British trad-jazz musicians who came to the fore in the 1950s—article by Anne Sharpley, published in the Evening Express (Aberdeen, Aberdeenshire, Scotland) of Friday 12^th August 1960:

Humphrey Lyttelton was one of the first [British trad-jazz musicians].
That inevitable “Old-Etonian-viscount’s-nephew-turned-trumpeter” appendage to his name tends to disguise the fact that his was not quite such a solitary repudiation of background as it seemed.
In fact it was the public schools that provided trad with its first wave of talent.
Of these the most eccentric was one “Mick” Mulligan—who if he did not coin the word “raver” certainly had the whole thing grow up round him.
[…]
“Raving” meant originally just playing instruments—and to hear Mick and George Melly, his blues singer, and the New Magnolia Jazz Band, one would never have doubted its aptness.
A language grew up—known only to a few initiates.
First to be labelled were the young men from public schools—rather the sort of young men they might have been. They were labelled “Hooray Henries” by James Godbolt, Mulligan’s agent, from a Damon Runyan story.
They had to have their female counterpart and these were first “Hooray Henriettas” and then finally “’etties.” Their un-hooray equivalents were “’erbs” and “Emmas.”
To label someone a hooray was not simply a question of spotting an upper-class accent; it had to be combined with a certain obtuseness, although it is worth noting in the matter of accents that all of the ex-public school contingent (including Lyttleton [sic] and Chris Barber) cultivated a neutral accent.