Here is what the official website of the Oscars says:
Officially named the Academy Award of Merit, the statuette is better known by its nickname, Oscar. While the origins of the moniker aren’t clear, a popular story has it that upon seeing the trophy for the first time, Academy librarian (and eventual executive director) Margaret Herrick remarked that it resembled her Uncle Oscar. The Academy didn’t adopt the nickname officially until 1939, but it was widely known enough by 1934 that Hollywood columnist Sidney Skolsky used it in a piece referring to Katharine Hepburn’s first Best Actress win.
And Wikipedia says:
The origin of the name Oscar is disputed. One biography of Bette Davis claims that she named the Oscar after her first husband, band leader Harmon Oscar Nelson; one of the earliest mentions in print of the term Oscar dates back to a Time magazine article about the 1934 6th Academy Awards. Walt Disney is also quoted as thanking the Academy for his Oscar as early as 1932. Another claimed origin is that the Academy's Executive Secretary, Margaret Herrick, first saw the award in 1931 and made reference to the statuette's reminding her of her "Uncle Oscar" (a nickname for her cousin Oscar Pierce). Columnist Sidney Skolsky was present during Herrick's naming and seized the name in his byline, "Employees have affectionately dubbed their famous statuette 'Oscar'". The trophy was officially dubbed the "Oscar" in 1939 by the Academy of Motion Picture Arts and Sciences.
So, no clear answer, support for the story you remember, but for others too. I can imagine a recipient of a statue award might give their particular statue a name to be funny, and then a friend of theirs who won the next year might say "now I have an Oscar of my own" and in no time that "in crowd" all call their statue (not the award, just the physical statue) the same thing. In a circumstance like that, exactly who nicknamed it first and why might end up never being correctly captured or recorded.