In Frankenstein, how does Elizabeth come to live with Frankenstein?
In Chapter 1 of Frankenstein, Victor writes:
So, Elizabeth was adopted by Victor as part of his traveling in Italy with his wife Caroline. Interestingly, Shelly's first draft was written that Elizabeth was actually Victor's cousin. In the 1818 originally published novel, Shelley writes her to be Victor’s cousin, the child of Alphonse Frankenstein’s sister. Later in the 1831 revision, Shelley changes her to be simply an unknown orphan girl. Apparently, there might have been outcries of incest, so Shelley backed away from the "kissing cousins" idea.
When my father returned from Milan, he found playing with me in the hall of our villa a child fairer than pictured cherub--a creature who seemed to shed radiance from her looks, and whose form and motions were lighter than the chamois of the hills. The apparition was soon explained. With his permission my mother prevailed on her rustic guardians to yield their charge to her. They were fond of the sweet orphan. Her presence had seemed a blessing to them; but it would be unfair to her to keep her in poverty and want, when Providence afforded her such powerful protection. They consulted their village priest, and the result was that Elizabeth Lavenza became the inmate of my parents' house--my more than sister the beautiful and adored companion of all my occupations and my pleasures.
Elizabeth is a young aristocrat whose mother was an Italian woman of means that was taken to prison. Due to this, Elizabeth was sent to live with a family who was asked to take her in. It was actually Frankenstein's mother who set her eyes on Elizabeth so that the girl could have the opportunity to be raised in an environment more worthy of her status. This is how Elizabeth landed in the Frankenstein household.