This is where taking note of the very first verse Mark writes (in his gospel account) throws some light on why Mark wrote what he did in 6:3.
"The beginning [arche, Heb. 6:1] of the gospel of Jesus Christ, the
Son of God." (Mark 1:1 KJV)
At the outset, Mark's identification of Jesus Christ as the Son of God provides the basis for understanding all other statements made about him. Only once the reader has accepted this identification, will all other points about who Jesus really is, make spiritual sense.
Mark's focus, from the very first words of his book, is the Messenger of the Covenant, the ministry of the aphesis of sins. That is why it's important to see the point Mark makes in chapter 6 verses 1 & 2, prior to this cry of disbelief. Bear in mind that Christ was assigned to preach "To the Jew first", to his own people. Here is an explanation from this book:
"Notwithstanding the Jews being of the seed of Abraham; irrespective
of their springing from the stock of Israel; despite that they were
Jews, faithful to the throne of David: when Jesus went with the gospel
'to the Jew first', he found few enough of the children of promise
that could be counted for the seed, for all the multitudes that were
born after the flesh.
Never was this more conspicuous than when in 'his own country', Mk.
6:1. The irreconcilable difference between the children of promise and
the bondchildren became more evident than ever. Here appeared a divine
distinction that prevailed over all, and would endure world without
'Is this not the carpenter [the son of Mary]?' they cried. 'And they were offended at
him.' But Jesus said unto them, A prophet is not without honour, but
in his own country, and among his own kin, and in his own house, Mk.
6:3,4. 'And he marvelled because of their unbelief', Mk. 6:6. And as
it was with those in particular, so it happened with the Jews in
Nevertheless this gospel was 'to the Jew first', and thereunto was he
sent. And he called unto him the twelve, and, empowering them, sent
them forth into all the land, two by two. 'And they went out, and
preached that men should repent', Mk. 6:12" Mark, pp. 123-4, John
This quote shows why Christ's detractors stressed him being "son of Mary" - because they had already taken offense at him, they would not consider any link to the line of David. Had they seriously considered the possibility that he could be the foretold Messenger of the Covenant, they would have taken into account the significance of Joseph's lineage to the line of David.
So, the answer is that it is not Mark who is choosing to put a different stress on this by emphasiszng Mary (compared with Luke's stress on Jesus and Joseph). It is the disbelievers in Jesus being the Messenger of the Covenant who betray this by speaking of Mary, when they could have spoken of Joseph. Mark's stress through all of his writing is that this Jesus is the Christ, the Son of God.