What we know to be the Gospel of Mark is not signed, and neither is there any direct internal evidence linking the Gospel to any individual. However, does this mean the work is truly anonymous? That is, do we have no reasonable idea of who its author is? The answer to this question largely depends upon your personal bias. If one completely rejects early testimony of its authorship and clings to modern criticism stating the author is unknown, then for you the author cannot be known. However, if you are willing to accept ancient testimony as evidence of its authorship, then one can be reasonably certain that John Mark, Peter’s assistant, wrote the Gospel we know by his name.
Early in the 2nd century AD Papias (cir. 60-130), the bishop of Hierapolis, refers to one he calls “the elder” saying: “Mark became Peter’s interpreter and wrote accurately all that he remembered… of the things said and done by the Lord” [Papias 2.3]. Papias goes on to tell us that Mark is not an accurate chronological account but accurate in what he states, because not having been an eye-witness he didn’t know the order in which the things the Lord said and did actually occurred.
The Greek word Papias uses for “interpreter” is hermenutes, and the context of Papias’ statement probably means “translator” which is a possible meaning of the Greek. Indeed it is so used in the LXX at Genesis 42:23 where Joseph spoke with his brothers through a translator. The point is that, even if Peter knew Greek or Latin, Mark could probably translate from the Hebrew or Aramaic better than Peter and was so used.
Speaking of Jesus cir. 150 AD, Justin claims in his Dialogue with Tripho,
“It is said that he changed one of the apostles to Peter; and it is written in his memoirs that he changed the names of others, two brothers, the sons of Zebedee, to Boanerges, which means ‘sons of thunder’…” [Dialogue with Trypho, 106.3]
If Justin is referring to Peter’s memoirs then this might be a reference to the Gospel of Mark, because only in it are the sons of Zebedee called the “sons of thunder.”
Another 2nd century AD source showing Mark wrote out Peter’s Gospel is Irenaeus (130-200):
“When Peter and Paul were preaching the Gospel in Rome and founding the church there… after their departure, Mark, Peter’s disciple, delivered to us in writing the substance of Peter’s preaching.” [Against Heresies 3.1.1]
Writing in his Hypotyposeis Clement of Alexandria (cir. 150-215 AD) tells us that he learned from “the elders from the beginning” that:
When, by the Spirit, Peter had publicly proclaimed the gospel in Rome, his many hearers urged Mark, as one who had followed him for years and remembered what was said, to put it all in writing. This he did and gave copies to all who asked. When Peter learned of it, he neither objected nor promoted it” [cited in EUSEBIUS: Ecclesiastical Histories, 6.14].
Tertullian, moreover, says of the Gospel of Mark,
“The same authority of the apostolic churches will afford evidence to the other Gospels also, which we possess equally through their means, and according to their usage — I mean the Gospels of John and Matthew — whilst that which Mark published may be affirmed to be Peter’s whose interpreter Mark was. For even Luke’s form of the Gospel men usually ascribe to Paul. And it may well seem that the works which disciples publish belong to their masters.” [TERTULLIAN: Against Marcion; 4.5]
In the Muratorian Fragment dated cir. 170 AD we find this:
 . . . But he was present among them, and so he put [the facts down in his Gospel.]  The third book of the Gospel [is that] according to Luke.  Luke, “the” physician,  after the ascension of Christ,  when Paul had taken him with him as a companion of his traveling,  [and after he had made] an investigation, wrote in his own name —  but neither did he see the Lord in the flesh —  and thus, as he was able to investigate,  so he also begins to tell the story [starting] from the nativity of John.
In the first line above, it probably refers to Mark, since the context has to do with the Gospel accounts, and may be considered to mean his presence among Peter’s discourses, if we take the previous evidence into consideration.
Admittedly, some scholars may claim that all this evidence is dependent upon Papias. However, this is not necessarily the case, and the burden of proof would be upon those who make such a claim. Moreover, Papias was held by some to be ignorant. If this is true, why would he be so well known throughout the Church that he is quoted by so many well educated Christian scholars?