By almost any measure, she was a "success."
She was part of a literary and intellectual dynasty, so to speak. Her father was a philosopher by the name of William Godwin, and her mother was a famous writer and women's rights advocate who authored the seminal work A Vindication of the Rights of Woman. She was also married to the famous British Romantic poet Percy Bysshe Shelley, whom she began seeing while he was married to his first wife.
Mary Shelley was certainly known and read and taken seriously as a writer during her lifetime. The first edition of Frankenstein came out in 1818, but it did not carry her name. Subsequent editions of the novel did carry her name. Shortly after the first edition of the novel came out, a play adaption was staged, and the story became more popular. Gothic novels were popular in the early nineteenth century, and many people consider Frankenstein to be the first science fiction novel.
Mary Shelley made money from Frankenstein and her other published works, but it is mainly for Frankenstein, as well as her marriage to Percy, that she is remembered today. She was certainly not a rich writer in her time, but it was difficult to earn a fortune from writing in those days, as publishers only paid a lump sum at the time of publication, as royalties were not a thing.