Lapide offers this:
Simon Bar-jona. For the father of Simon Peter was called Johanna, that is John, as is plain from S. John xxi. 15, meaning “God hath given: or God hath pitied: or the gift of God, from ‘Ia’ which is contracted from Jehovah, and ‘chanan,’ that is, he hath visited, he hath given.” Peter, then, was the son of John, or the grace of God, because he was most pleasing to God, and full of His grace. S. Chrysostom observes, that Christ gave the addition “Bar-jona,” not only according to the Hebrew custom, which always adds the name of the father to the children, but with a special reference to Peter’s answer, as though Christ confirmed it and said, “Thou hast spoken truly, 0 Peter, that I am the Son of God, for as thou art the son of Jona, a man from a man, according to natural generation, so am I the Son of God the Father, but begotten of Him from eternity—God of God, of one substance and Godhead with Him.” Symbolically Jona, that is “a dove,” is the emblem of the Holy Ghost, who in the form of a dove came down upon Christ. In this place also he descended upon Peter, and revealed to him that Christ was verify and indeed the Son of God. Thus S. Jerome—“Peter obtains a name from his confession, because he had a revelation from the Holy Ghost, whose son he was to be called.” Bar-jona in our language signifies “the son of a dove.” “For flesh and blood hath not revealed it unto thee”—that is, not earthly parents nor friends nor any man who consists of flesh and blood has revealed unto thee that I am the Son of God—forasmuch as this knowledge far transcends all nature, and the natural knowledge of all men, but My Heavenly Father hath made it known to thee by the illumination of His grace. “What flesh and blood could not reveal, has been revealed by the grace of the Holy Ghost,” saith S. Jerome. By flesh, S. Hilary understands the bodily eyes of S. Peter, for they had told him that Christ was a man, but the revelation of the Father alone had made known to him that He was God.