The name was first publicly used was in an article by Hollywood columnist about Katharine Hepburn's first Best Actress victory at the fifth annual ceremony in 1934. By 1939 the word was universally know - but the origins remain unclear.
One theory is that the name came from an early Academy director, Margaret Herrick, in 1931. According to this legend, Herrick thought that the statue looked like her Uncle Oscar. Another is that Bette Davies named it after her husband - but that was in 1936.
A good example of how a nickname can survive long after its source is forgotten.