According to the Oxford English Dictionary, the word "biff" was originally "imitative" slang used as:

An exclamation uttered when something strikes an object, or a sound imitative of such a blow.

The earliest instance they cite comes from the 1843 book Streaks of Squatter Life, and Far-West Scenes by John S. Robb (page 137):

"I hit him, biff, alongside of his smeller."

By the turn of the century, it was known in the UK. H.G. Wells used it in 1905 in his book Kipps: The Story of a Simple Soul:

"When I go to turn, if I don't remember, Bif!—and I'm in to something."

A couple other well-known authors who used it were D.H Lawrence in his 1934 book Modern Lover ("He...took the poker with satisfaction. Biff! A well-aimed blow.") and Arthur Miller in the 1949 play Death of a Salesman (the central conflict in the play is between Willy Loman, the salesman, and his oldest son Biff, who was a star high school football player who had dropped out before graduating).

"Biff" also had become a noun, meaning "a blow, whack", according to the OED. This is first found in Berre & Leland's 1889 Dictionary of Slang, Jargon and Cant, published in Edinburgh:

"Biff (Americanism), to give [one] a 'biff in the jaw'."

Green's Dictionary of Slang agrees, citing many more examples.

The nickname "Biff" appears to have been a loan from this previous American slang, used primarily to describe pugnacious types. Direct evidence of this can be found in the 1945 non-fiction book Larry Scott of the Sun by Edward Ford. Larry Scott, a reporter for the New York Sun newspaper, was assigned by his editor to interview a boxer by the name of Biff Kollock. Ford recounts that the editor had told Scott:

"...[T]his is a tough assignment. Kollock doesn't like newspapermen, not from papers on our side of the political fence, anyway. Try to make him talk but don't get too close. Kollock earned that nickname 'Biff' by bouncing right hooks off opponents' jaws. See what you can do."

A similar origin is implied in the review of the 1933 film (adapted from a stage play) One Sunday Afternoon, that appeared in Punch magazine. The film starred Gary Cooper as a dentist named Lucius Griffith "Biff" Grimes. The story is about Biff giving a dental treatment to a man who had been the successful rival for a girl they were both in love with during their youth. The story is mostly told in flashback, recounting their earlier rivalry. Biff was a working-class neighborhood tough ("a short tempered scrapper" as one online review puts it), while his rival was wealthy and suave. Punch describes Cooper's character with:

"[He's n]ot an ordinary dentist, I hasten to add: not neat and quiet and tactful, but a dentist all the same, even though the larger part of the story is devoted to the chrysalis stage when he was earning and justifying the nickname of Biff."

This is likely where Arthur Miller took the nickname from, too, as the argumentative, athletic ex-football player Biff Loman.

And these all agree with a recent baby name book which defines the origin of the name "Biff" with:

"Biff (American) A bully"

It appears to have been extrapolated somewhat beyond a nickname for a pugnacious person, to also describe a person who is tenacious.

Why it became popular as a nickname in the early to mid-20th Century and not earlier or later, maybe someone else can answer that, but generally speaking, names go in and out of style all the time. There are a lot of sports nicknames that showed up in that period ("Whitey", "Rocky", "Sparky", "Pop", "Pee Wee", "Kid", etc.) that are no longer common, or at least not as common as they once were. Some of them, like Biff and Rocky, crossed over into actual given names before largely falling out of fashion.