As far as the NT goes, it looks like he is only mentioned in Acts, and as far as Acts goes I bring some presumptions to the table. For reasons I've given on other threads and don't feel the need to get into now, I suspect it was written by Paul's (and later Josephus') patron Epaphroditus and that he was executed by Domitian c. 95 CE and that Acts was thus written no later than c. 95 CE.
And despite thinking that Acts is pro-Pauline and has an agenda of smoothing things over between Paul and Jewish Christian leaders, and despite also thinking that it uses (and distorts) Josephus, and setting aside that it makes up dialogue and uses what I call "special effects" (as more or less all ancient writings do), I think the author of Acts was in a position to know some things about early Christians and their writings, and it makes me wonder if there could thus be something to what they say about John Mark.
I also bring what Papias says about the gospel of Mark being written by a follower of Peter to the table, since Papias (who I date c. 100 CE) was also in a position to know some things about early Christians and their writings, and I take what he says about the gospel of Mark seriously.
I am also open to the possibility that the Mark in Philemon 1:24 could be Papias' Mark and/or John Mark, but I need to give that more thought.
And while I gather that the identity of John Mark with the author of the gospel of Mark (and/or Paul's Mark) is later patristic speculation, not knowing where else to start, I thought I'd look at what Acts says about John Mark with that idea in mind and see what I think about it.
Right off the bat Acts 15:36-40 jumps out at me:
Some time later Paul said to Barnabas, “Let us go back and visit the believers in all the towns where we preached the word of the Lord and see how they are doing.” Barnabas wanted to take John, also called Mark, with them, but Paul did not think it wise to take him, because he had deserted them in Pamphylia and had not continued with them in the work. They had such a sharp disagreement that they parted company. Barnabas took Mark and sailed for Cyprus, but Paul chose Silas and left ...
This reminds me of the what Paul says in Gal. 2:11-13 about Barnabas joining Peter when he and other Jews separated from Gentiles in Antioch:
When Cephas came to Antioch, I opposed him to his face, because he stood condemned. For before certain men came from James, he used to eat with the Gentiles. But when they arrived, he began to draw back and separate himself from the Gentiles because he was afraid of those who belonged to the circumcision group. The other Jews joined him in his hypocrisy, so that by their hypocrisy even Barnabas was led astray.
So here in Acts we have a guy named John Mark who followed Barnabas and not Paul, and in Galatians Barnabas sided with Peter and not Paul, which fits the idea that John Mark could be the Mark that Papias says was a follower of Peter. Hm.