In John 1:42 we read
He brought him to Jesus. Jesus looked at him and said, “You are Simon the son of John. You shall be called Cephas” (which means Peter).
While in Matthew 16:17 Jesus said to him “Blessed are you, Simon Bar-Jonah!”, meaning Simon son of Jonah.
Later, in John 21, Jesus repeated calls him “Simon, son of John” (John 21:15-17).
So his father’s name was either John or Jonah. Possibly, the Hebrew form of the name was Yohanan and Jonah is an Aramaic abbreviation, while John is from the Greek form of the same name.
We know nothing else about him, and there is nothing to suggest he can be identified with any other John in the New Testament.
sometimes “son of” is used to refer to descendant of… is it possible that the wavering disciple Peter was a descendant of Jonah the prophet? after his big fish encounter his descendants probably embraced their inner mariner so as not to be on God’s bad side in the future generations.