Since I made a statement in another post here that I believe Mark to be Peter's son, I will explain.
In Acts 12:12-17, Peter had just escaped from jail. He went to a house belonging to someone named Mary, but who is not further identified except as Mark's mother. Rhoda answered - sort of. Now many English translations are fond of identifying Rhoda as a servant. But the Greek does not require it. Indeed, KJV simply calls her a "damsel." Peter and Rhoda had a conversation through the closed door. Peter might have identified himself, but the text is clear to say that Rhoda knew it was Peter by his voice.
Now let's think on this. A single-digit age girl, with no access to CNN, recognizing the voice of a male celebrity. Add to this the delirium of her reaction to hearing Peter. She plumb forgot to open the door for him. This hardly qualifies as proof, but can anyone come up with a better explanation than Daddy came home safely? Rhoda's father, who had been dead to her, has been risen from the dead!
If such a young girl felt comfortable answering the door, there is a good likelihood (though not a certainty) that Rhoda lived there, and was the daughter of Mary. Since Mary was the mother of both Rhoda and Mark, if Peter was Rhoda's father, then Peter was also Mark's father. A variation on this is that Mark was Mary's son from a previous marriage. But those cases are more common today than they were then.
Pushing the other way:
1. Wouldn't it have been more natural for Luke to say it was Peter's house? Answer: It might not have been Peter's house. Of all people, why did he pick Mary's house to find refuge? He clearly went to a place where the people there could be trusted. We know that Peter was married when he received his call, and that he accepted the call without discussing it with his family. Which is more likely? That the married Peter went to a went to a woman's house where scandal could be reported - whether or not it happened? or that Mary was Peter's wife, who could not meet the rigors of being an Apostle's wife, and through an amicable separation (not divorce) Peter established her independence for her?
Actually, it is curious that Mary owned a home at all. Female-owned property in Judea was governed by Numbers 27 and 36. In order for a female to own property she had to inherit it, and then only if she had to have no brothers and have a father who died. But when she marries, assuming she marries intertribally and inter-family, then the property gets transferred to her husband. (If she marries extra-tribally, then she forfeits her inheritance, in order that a tribal map of Israel not look like an explosion at the pixel factory after many generations.) In other words, barring a very unusual circumstance, there was no lawful way for a married woman to own property. Since Mary was Mark's mother, we will presume she was married.
[As an aside, an extension of this doctrine is the reason why Luke records Joseph as the full son, and not the son-in-law of Jacob in Luke 3.]
2. Colossians 4:10 identifies Mark as the cousin of Barnabas. Wouldn't the more prominent reference be "Mark the son of Peter?" Answer. In an ideal world, yes it would. But it is not at all unusual for great leaders of the faith to have difficulty getting along. The latest high-visibility skirmish is probably Bob Jones, Sr. and Billy Graham. Graham had been a student at Bob Jones University, and had announced his intent to withdraw and start an evangelism ministry. At first, Jones disapproved on the grounds that Graham was not properly trained. Then later, the complaint was changed to one that Graham was preaching heresy. I could be wrong, but I don't think the rift was ever fully resolved.
John Calvin and Martin Luther had the world focused on the Gospel for several decades. But their failure to agree on the presence of God and the communion elements prevented them from teaming up. Oh, what could have been....
Likewise Peter was not very happy with Paul for correcting him publicly (Galatians 2:11-14). And 2 Peter 3:15 is a possible swipe at Paul (though the term "beloved" needs to sift through the filter of inerrancy). It doesn't stretch my imagination to think that Paul truthfully resorted to a secondary relationship for Mark in order to avoid mentioning Peter.