Remittance Man

In essence, a young man is requested to travel -- anywhere -- and is provided with a reasonable amount of money as long as he stays away from home.

This may be a happy solution -- if the money arrives in the proper place at the proper time.

—Alexei Panshin, Star Well

"That fop with the English Accent." An upper-crust younger son of an English lord with no prospect of inheriting, sent off to the Americas to get him out of the way. Usually given a small allowance (the "remittance"), that isn't sufficient to support him in the way he is accustomed. He also has an ingrained aversion to "working in trade", and he's not used to manual labor.

In Westerns, he's often connected somehow to the Cattle Baron; he may be the money-man or at least represent "the money", or be the "manager". Could also be a drunken wastrel with no visible means of support. In Canadian versions of the Old West he might be the local Mountie or the local criminal.

In Genteel Interbellum Setting mysteries, he's usually back from Australia or South Africa (occasionally South America, the US or Canada) and most of the family would prefer that he'd stayed gone.

Usually only appears in fairly realistic Westerns, except in Canadian versions where he's a stock character. Permits the inclusion of a different version of the City Slicker type, more "civilized", more condescending, and generally just as incompetent.

Sometimes in other settings (like the South Pacific) where any European is likely an outcast, with many of the same tropes still applying.

If he's smart, he'll almost certainly overlap with the Upper Class Wit.

Examples of Remittance Man include:

Comic Books

One shows up in The Tenderfoot, though he doesn't receive an allowance.

Film

English Bob (played by Richard Harris) in Unforgiven may or may not be a real Remittance Man. But he certainly acts like one (possibly as protective coloration, to intimidate people from bushwhacking him).
In a rare case of the American counterpart to the Mountie version, there is Sheriff John T. Langston, played by John Cleese in the 1985 film Silverado.
The unnamed Englishman in the Canadian short Wild Life or Une Vie Sauvage, the film starts out satirizing this phenomenon but takes a decidedly melancholy turn

Literature

"Ginger Ted" of Somerset Maugham's story "The Vessel of Wrath" (filmed as The Beachcomber) is explicitly described as one of these. He's a drunken lout who periodically receives sums to keep him from leaving the South Sea island where he resides. Despite his slovenly appearance, he sometimes evidences a high level of education.
In Alfred Bester's 5,271,009, the alien who helps the protagonist describes himself as a remittance man.
A sort of truth-in-television example is Frank Dickens, used in the novel (and Flashman pastiche) Dickens of the Mounted. Frank was the wastrel son of Charles Dickens and became a member of the Mounted Police in Canada.
A Robert Louis Stevenson novella The Beach of Falesa has one in the character Case, although he's more competent (and malevolent) than most. The tale is set on a fictional island in the South Pacific and Case is along with protagonist, among the few white traders who live there and is a ruthless and amoral schemer. The protagonist describes how Case would sometimes discourse in an intelligent, cultured way and you can kind of tell from his speech that he was once a toff (i.e. calling the protagonist "old boy"). There's an amusing detail that while the other whites mispronounce the name of a French priest Galuchet as "Galoshes", Case can pronounce it correctly. Case also qualifies as a Mister Danger type, since he uses magic tricks and some technology to trick the natives into thinking he has demonic powers, allowing him to have a great influence over them.
A couple of Bertie Wooster's friends. In "Jeeves and the Hard-Boiled Egg," he lends his apartment to Bicky Bickersteth (of the "wastrel" variety, naturally, living in a boarding house in New York when he's supposed to be farming in Colorado) so he can make his uncle think he's doing well in America. It works too well and the uncle decides to withdraw Bicky's allowance, since he clearly doesn't need it.
In The Great Gatsby, the first time Nick goes to one of Gatsby's parties, he notices several young Englishmen among the guests, "all well dressed, all looking a little hungry, and all talking in low, earnest voices to solid and prosperous Americans."
At the end of The Way We Live Now by Anthony Trollope, the nasty cad Felix has racked up some very high gambling debts. In exchange for those being covered, he's sent to an enclave of British clergy in Germany and receives support there, and is basically told not to come back to England.
Mym of the Incarnations of Immortality series is a non-European variant of this trope, a prince of India who didn't fit in with the royal court because he can't talk without severe stuttering. He travels around unrecognized with a circus, within the borders of India. Since he is the second son, the royal court's policy is to tolerate his runaway lifestyle - until Mym hears the news that his brother has died in a war, and the court, who has been secretly tracking his whereabouts all along, will begin insisting that Mym shall come back to the palace and live the lifestyle appropriate to the heir to the throne.
British secret agent Captain Patrick Reeder pretends to be one in The Remittance Kid by J. T. Edson.
Alexei Panshin's hero Anthony Villiers, Viscount Charteris, is one of these (and the subject of the page quote) in a science fiction series (cut short after three books) consisting of Star Well, The Thurb Revolution, and Masque World. A supplement for Traveller did a write-up of Tony as an example; it mentioned that he's at ease in any social situation. He's very definitely the smart variety, and to some degree, the books (especially Star Well) shade into detective/spy stories.

At first glance, Villiers was not impressive. He was young, mild, well-mannered, well-dressed, and essentially ordinary. At second glance, he had considerable charm. At third glance, he was obviously the sort of person around whom things happen, and why hadn't you noticed that before? We won't even mention his hidden virtues. ... Whimsical curiosity was one of his major failings of character.

In H. Beam Piper's Four-Day Planet, people believe "Bish" Ware is a remittance man; it's thought he embarrassed some wealthy organization enough that they'll pay him to stay on the title world. "Bish" is short for bishop, because he looks like a stereotype of a fairly senior clergyman, and the mildest suggestion as to what crime he committed to be exiled is that he stole the funds intended for constructing a cathedral. He's actually a high-ranking secret agent, tracking down a particularly vile criminal.

Music

Jimmy Buffett's "Remittance Man":

Black sheep of the family clan
Broke too many rules along the way

Tabletop Games

A character type in the Traveller roleplaying game. (Though it's a science fiction setting, there are plenty of useless nobility around.) Perfect for the player who wants an eclectic skill set, no fixed responsibilities, and a good motivation for adventuring (i.e., get money).

Real Life

Thicker on the ground than gophers in pre-World War I Calgary, which probably explains why the trope is more common in Canadian shows. One old apocryphal joke had a local lawyer writing the noble father of a remittance man who was convicted of murder and hanged: "I regret to inform Your Lordship that your son has died. He was participating in a public function when the platform gave way."