Why have she changed her name to Lucille?
At first, she didn’t changed her name to “Lucille”: Lucille became a model and later she begin a career on Broadway, it was in 1929, Lucille begin to using the stage name “Daine Belmont”. The name “Daine Belmond” sounded more classic and prettier than “Lucille”. She thought it would help her career, but it didn’t. So she named herself “Lucille” again, her own name.
Why the name Lucy?
When Lucille was younger, she didn’t like to be called “Lucy”. The name “Lucille” was more elegant and classier than “Lucy”. But that changed when she met Desi. After the first night together, Desi started to call her, Lucy. He ‘hate’ the name “Lucille”, because he said: that other men had called her Lucille, Lucy was his alone. From that moment she didn’t mind being called “Lucy”, she even loved it. So you see what love can do.
So here is a short summary: The name “Daine Belmont” was her stage name, in her younger years; “Lucy” is the name Desi called her; “Lucille” Ball is her real name.
If you can see I call Lucille mostly “Lucy” because I love that name. Especially because Desi called her so.