Why is the Gospel of Luke the only Gospel to claim Jesus was John the Baptist's cousin? Why is Luke's Gospel the only of the four Gospels to make the assertion that Jesus was the cousin of John the Baptist?
Luke sought eyewitness testimony for his gospel. The birth narrative and some other portions seem to be told from the viewpoint of Mary the mother of Jesus.

While it is somewhat speculative, it seems reasonable that Mary would be concerned about relationships within her own extended family much more than the majority of the early Christians. It would appear that Luke either interviewed Mary personally or talked with someone who knew her very well.

According to this article, there is every reason to believe that Mary did indeed pass on her story in detail, and we know that Luke was interested in exactly the kind of details that could only have come through her testimony.


  
  Did Mary provide some traditions for the Infancy narrative?
  Why would she not tell her story? It would be rather odd if she had remained silent! She was part of the Jerusalem church (Acts 1:14, with Jesus’ half-brothers), and Luke had knowledge of this lead church. He somehow collected Mary’s story and put it in his Gospel, just as he incorporated Cleopas’ and Zacchaeus’ eyewitness testimonies. The Infancy narratives in both Matthew and Luke have a huge body of scholarly literature behind them, so in an article like this one I would not dare to figure out which is the core of the story that Mary passed on, though these verses in Luke seem like good candidates for further study: 1:26-56; 2:4-7, 19; 21-49.