The most famous nickname in film has at least four people claiming it, and the evidence supports none of them cleanly.
The statuette has been given out since 1928. By the early 1930s, people in Hollywood were calling it Oscar. Etymonline dates the nickname to 1933 and calls the Herrick story the common explanation, adding that other origin stories exist.
Margaret Herrick. She was the Academy's librarian and later its executive director. The story goes that in 1931 she looked at a statuette and said it reminded her of her Uncle Oscar. Oscar Pierce was a wheat and fruit grower, and one account says he was a cousin of her mother. Wikipedia says she "may have said" this, and the only corroboration is a 1938 clipping.
Sidney Skolsky. The Hollywood columnist said in his memoirs that he coined it in 1934, partly to poke fun at the Academy's pretension, and recalled a vaudeville joke. The Academy credits his March 16, 1934 column as the first confirmed newspaper reference. But the column already says the statuettes are called Oscars, which suggests the word was in use.
Bette Davis. She wrote that she named it for her first husband, Harmon Oscar Nelson, after winning in 1936. The nickname was already in use by then, and Skolsky says she later dropped the claim.
Eleanore Lilleberg. Bruce Davis, a former Academy executive director, argues for this Academy secretary. He cites her brother Einar's autobiography, which mentions a Norwegian army veteran named Oscar they knew in Chicago. Davis says credit "should almost certainly belong to" her. Davis himself acknowledges the evidence is not conclusive.
Earlier print. Researcher Waldemar Dalenogare Neto found what Wikipedia reports as the probable first public mention. It is in Relman Morin's December 5, 1933 "Cinematters" column in the Los Angeles Evening Post-Record. Wikipedia's Academy Awards article shortens the paper's name to Evening Record. That predates Skolsky's column by more than three months.
A popular idea that the name honors King Oscar II of Norway is dismissed by Davis as a misconception.
Biography.com says the true origin may never be known. A reasonable summary is that the name was already circulating in 1933, and each claimant may have been repeating something already in the air.