I am aware that there are a number of early church fathers who made this connection (e.g. see Eusebius and others including Irenaeus, Justin, Clement, Origen, and more cited in this blog post). Is there also material internal to Mark's gospel to suggest that it reflects the memoirs of Peter?
There is no conclusive internal evidence but there are plenty of pointers that lend themselves to the conclusion that Peter is in some way the source, for example this blog post lists some examples:
Peter is the first and last named disciple in Mark (1:16; 16:7).
Peter is mentioned more than any other disciple in Mark.
Peter appears in some of the most important scenes in Mark: the calling of the first disciples (1:16-20), the confession of Jesus as Messiah (8:27-30), the transfiguration (9:2-8), the prayer in Gethsemane (14:32-42), and in the concluding scene alluding to future appearances of Jesus (16:7).
Of the four gospels, Mark has the highest percentage of references to boats, the Sea of Galilee, and fishing. Peter apparently was a fisherman who worked on the Sea of Galilee (1:16).
There is the curious story of the healing of Peter’s mother-in-law (1:29-31), which seems to include personal details related to Peter.
Of these I give the last the most weight.
Also, only Peter, James and John were present at the transfiguration, so it is likely that one of them passed on the account to Mark:
2And after six days Jesus took with him Peter and James and John, and led them up a high mountain by themselves. And he was transfigured before them, 3and his clothes became radiant, intensely white, as no one on earth could bleach them. 4And there appeared to them Elijah with Moses, and they were talking with Jesus. 5And Peter said to Jesus, “Rabbi, it is good that we are here. Let us make three tents, one for you and one for Moses and one for Elijah.” 6For he did not know what to say, for they were terrified. 7And a cloud overshadowed them, and a voice came out of the cloud, “This is my beloved Son; listen to him.” 8And suddenly, looking around, they no longer saw anyone with them but Jesus only. Mark 9, ESV
This is very similar to Peter's account in 2 Peter:
17For when he received honor and glory from God the Father, and the voice was borne to him by the Majestic Glory, “This is my beloved Son, with whom I am well pleased,” 18we ourselves heard this very voice borne from heaven, for we were with him on the holy mountain. 2 Peter 1, ESV
So, Is there also material internal to Mark's gospel to suggest that it reflects the memoirs of Peter?
Yes, but the key word here is "suggest". There is no direct proof but also no strong reason to suggest otherwise, which is important too.
Here is the various internal evidence that I am aware of, as well as evidence that indicates a "persecuted audience" which fits the idea of this being written after Peter's death at the hands of Nero.
Evidence That could Indicate Peter as an Original Source
It is possible to see connections in the simple, quick and unpolished nature of this gospel and in Peter’s brash personality. This comes out through a linguistic analysis of the text: (1) Mark's most famous phrase: “and immediately” (gk. kai euthus) 37 times, 11 x's in ch.1 (2) Historical Present used over 150 times, creating a vivid atmosphere (3) Constant use of Parataxis: the joining together of sentences with conjunctions.
There are certain details that read like the telling of an eyewitness account (1:33,2:4; 4:37; 5:5; 6:40; 9:36; 10:16,21). Also the words Jesus originally spoke in Aramaic are recorded, as if the source of the gospel remembered them vividly (3:17; 5:41; 7:11,34; 14:36; 15:22,34)
Peter’s proclamation patterns in Acts is very similar to the gospel of Mark.
Simon of Cyrene who carries the cross for Jesus is stated to be the father of Alexander and Rufus (15:21). It is possible that Rufus was a roman believer and is the same one mentioned by Paul in his final greetings in Rom. 16:13. If this man was not known by the OR, it would not really make sense to record his son’s names. This is simply interesting speculation and could serve to connect the book to Rome, as church tradition suggests.
Internal evidence indicating a setting of persecution:
The brevity of the text could point to the fact that this was read by a persecuted church not a “philosophical or pondering church” (use of word “immediately”, emphasis of teachings over stories, cuts straight to the chase)
It looks like the original ending of the book is a group of disciples “afraid” which would be a very relatable experience for the OR
The moment of cruciality in the middle of the book when the messianic secret is unveiled is paired with very clear words regarding “losing one’s life for the sake of Jesus and the gospels” (mk. 8:34-35)
Passion Predictions show that the cross is a part of the Christian way, not a mistake (8:31-32, 9:9-10; 9:30-32, 10:32-34). Mark devotes a greater proportion of space to the passion narrative then any of the other gospels.
The crux of the paradoxical messianic secret is that the messiah is a suffering servant who will conquer through the cross and his disciples must too follow in the way of the cross Repeated Idea: Those who are last of all are first (9:35, 10:29-31, 42-45) Warning of Persecution: 13:11-13
There is really no evidence internal to Mark's Gospel to point to Peter having provided material for Mark's Gospel. Much of the basis for this assumption is that the gospel was written by Paul's companion, Mark, who was also Peter's son. However, the gospel was originally anonymous and there is nothing in the gospel even to link it to Mark.
It appears likely that Papias, early in the second century, was the first to propose Mark as the author of this gospel, and to suggest that he acted as the interpreter for Peter. The fourth-century Church Father, Eusebius (Ecclesiastical History, 3.39), quotes Papius, who is apparently referring to an earlier source, James the Elder:
The Elder used to say: Mark, in his capacity as Peter’s interpreter, wrote down accurately as many things as he recalled from memory—though not in an ordered form—of the things either said or done by the Lord.
Eusebius' history, if correct, at least in attributing to Papias the connection of Mark to the gospel, points to Papias as providing the link to Peter, with Mark as his secretary. Eusebius went on to say that Papias also cited 1 Peter 5:13, where Peter speaks of "my son Mark," as evidence that Mark and Peter had a close relationship. Doubts over the authorship of 1 Peter make the epistle an unreliable source, but Papias was not to know this.
If the testimony of Papias is unreliable in respect to the authorship of Mark's Gospel, we ought to look more closely inside the gospel for evidence of Peter's influence. There is nothing of a personal nature that would point to Peter as the source or the author to be Peter's son. Peter is always mentioned in the third person, and the healing of Peter's mother-in-law is treated dispassionately, with no suggestion of fear for her well-being or relief after her cure. Furthermore, Peter is frequently portrayed as obtuse and lacking in understanding, militating against this coming from Peter. The important account of Peter denying Jesus three times could hardly have originated with him.
On the other hand, if we find evidence that the author of Mark used sources other than Peter, this would lead to the conclusion that Peter was not the source for this gospel. , Raymond E. Brown says in An Introduction to the New Testament, page 160, that Mark seems to depend on traditions (and perhaps already shaped sources) received in Greek. He points out (page 162) that parallels have been detected between Mark and Paul's letter to the Romans, which was written somewhat earlier.