There are several interesting features about the NT data regarding Peter's names.
Matthew does not have a story about Jesus giving Peter his nickname, unless Matt. 16:18 serves that function (cf. 4:18; 10:2). But other than that single verse, Jesus in Matthew never calls him Peter, but only Simon, though he only calls him Simon a couple times (16:17; 17:25). Peter does not reappear in Matthew's Gospel after his denials in ch. 26. In other words, he does not feature in chs. 27-28 at all, except by implication as part of the Eleven disciples mentioned in 28:7, 8, 10 ("brethren"; or is this literally Jesus' brothers?), 16.
Mark never calls him Simon after he receives the name "Peter" (3:16) until Jesus finds him asleep in Gethsemane (14:37). The verse is very striking because the narrator uses the name Peter, and Jesus uses the name Simon, and these two words appear back-to-back in the Greek (and English).
He came and found them sleeping; and he said to Peter, "Simon, are you asleep? Could you not keep awake one hour?On the other hand, this is apparently related to the entire phenomenon of the name for Peter typically used by Jesus, which is actually (and counter-intuitively) Simon. In Mark, Jesus directly addresses Peter by name only once, here in Gethsemane, calling him Simon (14:37). Mark does report that it was Jesus who gave Peter his nickname (3:16). Also, in contrast to Matthew, Peter does make a brief appearance by name (in the mouths of the angels) following the resurrection (16:7).
Luke reports that Jesus gave Peter his nickname (6:14). After that, he is not called "Simon" again until Jesus says:
"Simon, Simon, listen! Satan has demanded to sift all of you like wheat" (22:31)But, after the resurrection, the disciples are still calling him Simon, when they say:
They were saying, "The Lord has risen indeed, and he has appeared to Simon!" (24:34)
Like Mark and unlike Matthew, Luke has Peter put in a brief named appearance following the resurrection (24:12, 34). Jesus addresses Peter by name as Simon (22:31) and as Peter (22:34). These are the only times in Luke that Jesus calls Peter by name, and they are both within three verses of each other, and they're two different names. And, as I said, the "Eleven" mention him as Simon in 24:34.
John is interesting, too. The references to Peter by his birth-name are most numerous in the Fourth Gospel, though almost always with the double name Simon Peter (see below). This is also the only Gospel that identifies the father of Judas Iscariot as Simon (6:71; 13:2, 26). I don't know if there is anything to that.
John alone among the Gospels actually preserves a story of Jesus giving Peter a nickname, and John uses the Aramaic form of the name, Cephas (1:42). Nevertheless, the rest of the Gospel uses the name Peter or Simon Peter. Jesus never again in John addresses Peter by his nickname. He addresses him as Simon a few times (1:42; 21:15, 16, 17). John uses the form Simon Peter much more than the other Gospels. This double name appears in John 17x, whereas he uses the single name Simon only 5x, four of which are in the mouth of Jesus (1:42; 21:15, 16, 17; the fifth occurrence is 1:41, before the receipt of the nickname), and the single name Peter 15x (1:44; 13:8, 37; 18:11, 16, 17, 18, 26, 27; 20:3, 4; 21:7, 17, 20, 21).
All four Gospels display in their narrative the effect of Jesus' nickname. Matthew twice mentions that Simon is also called Peter (4:18; 10:2), but he otherwise always calls him Peter, except in 16:16 when the narrator calls him Simon Peter, and twice Jesus calls him Simon (16:17; 17:25). Marks introduces Peter under the name Simon (1:16, 29, 30, 36; 3:16), but when he receives the nickname in 3:16, he is thereafter only Peter, until Jesus calls him Simon in Gethsemane (14:37). Luke calls him Simon several times when he first appears in the Gospel (4:38, 5:3, 4, 5, 8 ["Simon Peter"], 10; 6:14), but after he is named Peter in 6:14, he retains that name until Jesus refers to him as Simon in 22:31, just as the "Eleven" also do in 24:34. When Peter is introduced in John, it is as Simon Peter (1:40) or Simon (1:41), but Simon never appears again after Jesus gives him the name Peter in 1:42, except in the double name Simon Peter (17x, by the narrator) and in the mouth of Jesus (21:15, 16, 17). In none of the Gospels is he simply Peter until after the name change is reported.
Peter is the preferred way for the narrators to refer to him in each of the Gospels and in Acts. This seems to apply even to the Fourth Gospel, though the double name Simon Peter actually appears more often. This seems to be editorial, for whatever reason, but the narrator does also slip into calling him simply Peter quite a bit. The narrator never slips into calling him simply Simon after the name change. Simon Peter is also used in Matt. 16:16, where it seems to prepare the way for the two different uses of Peter's name by Jesus (Simon, 16:17; Peter, 16:18). When Luke uses the double name Simon Peter in 5:8, before his report of the name change in 6:14 and in the midst of an account in which he constantly uses the name Simon (4:38, 5:3, 4, 5, 10), it may have been a device to signal to the reader who Luke is talking about, assuming that Luke's readers may have been more familiar with the name Peter.
Thus, with this strong emphasis on the name Peter, it is striking when characters refer to Peter not by his nickname given by Jesus but by his birth name. Indeed, after giving Peter the nickname, Jesus more often refers to him as Simon (Mark 14:37; Matt. 16:17; 17:25; Luke 22:31; John 21:15, 16, 17) than as Peter (Matt. 16:18; Luke 21:34). Also, the "Eleven" refer to him as Simon rather than Peter in Luke 22:34. Even in Acts it seems that he is known as Simon, judging by the Cornelius story in which the phrase "Simon who is called Peter" is used multiple times (10:5, 18, 32; 11:13). And yet, in the same story, the voice in the dream tells "Peter"--calling him by that name--to kill and eat (10:13; cf. 11:7). If the narrator presents Peter as dreaming that God is calling him Peter, does that mean he thinks of himself as a person named Peter?
But then we have Paul, who almost always calls him Cephas. Presumably this means that the Christians in Corinth and Galatia knew Peter by the Aramaic form of his name. After all, Paul represents some Corinthians as declaring "I am of Cephas!" (1Cor. 1:12).
Paul's evidence suggests that Peter was known early--at least, outside of Palestine--by the Aramaic form of his nickname, and the Gospels suggest that by the time they were written, this was being replaced in the tradition with the Greek form. Does this mean that when Paul twice refers to him with the Greek form "Peter" in Gal. 2:7, 8, that this is somewhat innovative and unusual? If so, it would probably have been more rhetorically effective, if indeed Paul is contrasting Peter's nickname with his role in the Antioch affair (Gal. 2:11).