Here's a breakdown:
* Cio-Cio-San: This is the Italian pronunciation of her Japanese name "Chō-chō-san" (蝶々さん). "Chō-chō" means "butterfly" and "san" is an honorific, roughly equivalent to "Miss" or "Ms.". So, "Chō-chō-san" translates to "Miss Butterfly."
* Cho-cho-san: The spelling "Cho-cho-san" is a common mispronunciation of the Italian "Cio-Cio-San," reflecting the difficulty for some English speakers to pronounce the "Cio" sound.
While "Cho-cho-san" is not grammatically correct, it's widely understood as a reference to the character from the opera.