The Greek goddess who is the personification of the river Styx is Styx.
Styx is both a divine figure and one of the rivers of the Greek Underworld. In the usual genealogy, she is the daughter of the Titans Oceanus and Tethys, an Oceanid, and the wife of the Titan Pallas. Their children include Nike, the personification of victory, along with Kratos, Bia, and Zelus.
Styx gained special authority during the Titanomachy, the war between Zeus and the Titans. She was said to be the first deity to join Zeus, bringing her children with her. In return, Zeus made her water the basis of the gods’ most solemn oath. A deity who falsely swore by it faced a severe mythic punishment.
A common mix-up is treating Styx only as a river crossed by the dead. That image is especially prominent in later literature and popular culture; in Greek myth, Styx is also its divine personification. Ancient sources sometimes give her different parentage, but the identity of the goddess remains Styx.