The label 'lady in red' originated with a woman named Anna Sage who assisted the FBI in the capture of gangster John Dillinger in 1934. The oldest profession is prostitution and typically those women are called 'ladies of the night' but Anna became the 'lady in red' due to her outfit she was wearing the night Dillinger was caught, she was also a Madam of a brothel. Her outfit was actually an orange skirt and white top but the nickname was given nonetheless based off of her promise to wear red - even though she changed her mind about that. You can read all about it here.
How it came to be a saying today and what it actually means is a mystery but there is a song about it, sung by Chris De Burgh.