A New name = New Identity.
John 1:42 Then Andrew brought Simon to meet Jesus. Looking intently at Simon, Jesus said, “Your name is Simon, son of John—but you will be called Cephas” (which means “Peter”).
The introduction of Peter to Jesus was brief but direct. The simple pronouncement “ ‘You are Simon son of John. You will be called Cephas’ (which, when translated, is Peter)“ was really a diagnosis of Peter’s personality. Simon, or Simeon (cf. Acts 15:14), was the name of Jacob’s second oldest son (Gen 29:33), who, with his brother Levi (29:34), had ruthlessly avenged the violation of their sister by one of the Canaanite princes (Gen 34:25-31).
The rash and impulsive character of Simeon was mirrored in Simon, whose conduct as reported by all the Gospels that Simon reflects the same recklessness and tendency to violence (cf. John 18:10). Jesus accepted Simon as he was but promised that he should become Cephas, an Aramaic name, which, like the Greek “Peter,” means “a rock.” The development of Peter as recorded in this Gospel demonstrates the progress of that change. (Expositors Bible Commentary)
A new name means that a new identity, a new beginning.