Most people assume "scrooge" became a word the moment Dickens published A Christmas Carol in 1843. Dictionaries disagree about how fast it happened. Merriam-Webster gives 1899 as the first known use for a miserly person, and Etymonline says the common noun is attested by 1905.
Both dates are records of what editors have found, and they turn out to be late. Wordorigins.org quotes a Boston Christian Register of 19 October 1844 calling someone "a perfect Scrooge of a man," and says the generic sense caught on in less than a year. So the name did turn into shorthand almost at once.
Where Dickens got the name is less certain than the character's fame suggests. One suggestion is that the surname was partly inspired by "scrouge," an early-1800s word meaning to crowd or squeeze. Spelling variants such as "scrowge" and "scroodge" were also used for squeezing someone or crowding into their space. The verb is dated to roughly 1820 to 1830, which fits Dickens's era. Etymonline says older dictionaries link the name to an 18th-century variant of scrouge, and that the same word gave us scrounge. These are commentators' theories. Dickens left no account saying so. Wordorigins.org argues the name also evokes the slang screw for a miser, which was in print by 1825.
The first name has its own proposed meaning. Ebenezer means "stone of help," and it has been suggested that Dickens chose it to reflect the help Scrooge receives from three ghosts to change his life.
Then there is the gravestone story. It says Dickens saw a stone in Edinburgh's Canongate Kirkyard for a merchant named Ebenezer Lennox Scroggie. The stone supposedly called him a "meal man," meaning a grain merchant. Dickens is said to have misread it as "mean man." The tale is widely repeated, but Wikipedia calls it a probable hoax with no corroborating evidence. It adds that no one named Scroggie appears in the Edinburgh census returns of the period, and that the stone is now lost. A local Edinburgh guide site says no such stone survives and calls the story untrue.
So the safe version is this: Scrooge is an invented name, the word for a miser comes from the character, and the reason Dickens picked those particular sounds is unproven.
The character's other contribution to English is his catchphrase, "Bah! Humbug!", still used to dismiss Christmas cheer.