Sarah is the daughter of Norman Osborn and Gwen Stacy and her twin brother is Gabriel Stacy.
Before Gwen Stacy's death, she was seduced and impregnated by Norman Osborn. She gave birth to premature though fully grown twins while staying in Europe. Norman came and raised them, convincing them that Peter Parker was their father and that he had killed their mother as Spider-man. Their upbringing made the twins want revenge. Sarah and her twin brother Gabriel were born with the goblin powers inherited from Norman Osborn. Both twins aged at an accelerated rate because of these powers. Sarah, the smarter of the two twins, bears a striking resemblance to her mother. Sarah was saved by Spider-man twice: first when she fell off the bridge and second when Spider-man donated his blood to her after she suffered a bullet wound and her body rejected all other blood donations. Sarah shot at Gabriel (to protect a weakened Spider-Man), causing the Glider to blow up. He landed on a beach, with amnesia. He later returned home to Paris where Sarah secretly cared for him. While his sister left to be trained as an Interpol agent, Gabriel was last seen in a crazed state fleeing the Osborn estate. Sarah is currently allied with Interpol to find Gabriel.
During Brand New Day, it is believed to no longer exist as he was mentioned only once and not mentioned again. This is proven to be true, in one issue where Spider-Man states that Harry Osborn is Norman's "only son". It is mentioned by Norman that he cannot impregnate a woman who hasn't taken the Goblin serum; something that Gwen didn't do. When Molecule Man tortures Norman and the Dark Avengers; in Norman's memories one shows a pregnant Gwen Stacy being killed. This implies that Sins Past is not retconed. It is believed that when Norman stated that he could only impregnate a woman who has taken the Goblin Serum; he may have been referring he could but with massive side-effects, and since Lily Hollister had taken the Goblin Serum; her child my have a chance for normalcy. Yet, it also implies whether or not Sins Past has been retconned or not.
It is unknown who's last name she and her brother took; their mother's, their biological father's, or their alleged father's. So it is to assume that she and her brother took their mother's name.