As my mother was interested in opera I had imagined that she might be named after someone of Italian descent, perhaps even Bellini's Norma.
However, she once explained that the name came from a star of the early age of silent cinema: Norma Shearer (1902 – 1983). As an early feminist, my mother battled daily against the idea of being a silent woman, and it was perhaps the more confident version of Shearer that her parents might have being thinking of.
Shearer was a Canadian-American actress whose film career stretched from 1919 to 1942.
She was nominated five times for an Academy Award for acting. She won the Academy Award for Best Actress for The Divorcee (1930).
Shearer's roles were confident, perhaps even sexually liberated ingénues.
There were adaptations of Noël Coward, Eugene O'Neill, and William Shakespeare.
"the exemplar of sophisticated 1930s womanhood ... exploring love and sex with an honesty that would be considered frank by modern standards."
Shearer is celebrated as a feminist pioneer: "the first American film actress to make it chic and acceptable to be single and not a virgin on screen." (Mick LaSalle)
Sources
Wikipedia. https://en.wikipedia.org/wiki/Norma_Shearer
LaSalle, Mick. Complicated Women: Sex and Power in Pre-Code Hollywood. New York: St Martin's Press, 2000.