What was the origin of the phrase "15 minutes of fame"?

Solution 1:

If it's only similar that suits you, there are many references — a variety of sports reporters, for example, are and were big on the '[sometime] of fame' phrasing. Here's an early attestation from the Liverpool Evening Express, 06 September 1941 (paywalled):

This bespectacled Scot, typically dour, who talks through clenched teeth clutching his inevitable pipe, is fond of recounting the story of his 90 minutes of football fame.

MIRRLEES, OF FALKIRK.
Visiting a village near Glasgow, he "scrounged" a game with the local church eleven. Asked who he was, he gave his name, "Mirrlees, of Falkirk."

He never "got a look” at the ball throughout the game, as he seemed to be "policed" by at least six players all the time.

Reason for this respect and attention came after the game, when he learned he had been mistaken for a player from Falkirk, the famous Scottish First Division eleven!

What might be more interesting, and a closer match for the highly derivative phrase attributed to Warhol, was the renown of astronaut Alan B. Shepard, Jr., for a man-in-space (note the strong futuristic element) exploit in a Mercury 7 space capsule. Here's an excerpt from an 8 May 1961 story in the Philadelphia Daily News (Warhol's home state, seven years prior to the apocryphal aphorism's supposed first appearance):

Things were happening for Shepard almost as fast as they did last Friday when he flew 15 minutes into fame aboard "Freedom 7," a Mercury capsule ....