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:
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:
This is very similar to Peter's account in 2 Peter:
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
Internal evidence indicating a setting of persecution:
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:
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.