I'm pretty well read in this area, and I have never come across the theory that the Gospel of Mark was written by Peter's son (or even that Peter had a son, never mind his name). Where are you getting all of this?
Most of the associations between Mark and Peter come from Papias (whom we possess only in fragments). He makes no suggestion of a filial relationship between the author of the second gospel and Peter.
I Peter 5:13 identifies Mark as his son. It might just be a metaphor, but the text is clear enough that you should be aware of it.
I'm aware of the verse. I'm not aware of any scholar taking that meaning literally and concluding that 1 Peter (the author of which is unknown) is referring to Peter's biological son and further concluding that this son is the author of the Gospel of Mark (the author of which is also unknown).
I guess at this point, I'm legitimately asking who holds this view such that one can say "It is thought that . . ."
Well it was in the introduction of first bible I was given years ago - but that was 40 years ago and I can't find it anymore to give you more details. I've heard others say it - they might be a group repeating each other, but still that is two different groups.
What is the generally accepted meaning?
It could be affection and spiritual relationship. Or it could be a biological son but this Mark is not John Mark who is likely to be the evangelist. https://overviewbible.com/john-mark/
Please check https://overviewbible.com/john-mark/ even if he had a son called Mark that was not John Mark who is likely the evangelist. It's possible "son" here is meant spiritually.
Luke was not a disciple, but the rest of his works read like a scholar doing careful interviews of eye witnesses to find the truth.