Her name
I never saw the movie but saw the title listed in various sources and always had
trouble remembering if it was Culney Brown or Cluny Brown - and I can't promise that I will remember it right in the future.
No doubt there are scenes in the movie where the odd name is commented on and maybe explained.
Cluny is probably a place name that became a surname and then a surname that became used as a personal name.
There are places in Scotland named Cluny, from the Gaelic word for "meadow", in Aberdeenshire, Fife, Moray, and Western Highlands, as well as a Cluny in France.
https://en.wikipedia.org/wiki/Cluny_(disambiguation)
Cluny in France was part of the Duchy (not kingdom, nor county) of Burgundy and is now part of the department of Saone-et-Loire. It is famous for the Benedictine Abbey of Cluny which had the largest church in Europe from about 1100 to 1600 and was a leader in the Cluniac reforms of the catholic church.
https://en.wikipedia.org/wiki/Cluny_Abbey