We don’t actually know why they came to find him. This story shows up in three places (Mark 3: 31-35, Luke 8:19-21, and Matthew 12:46-50), and none of the evangelists mentions the reason why Mary and Jesus’ brothers (often understood to be half-brothers or cousins; see the New American Bible footnote for Mark 6:3) are seeking him out. Mark’s version does hint at some worry on the part of Jesus’ family; just prior to this passage, in 3:21, some unnamed relatives of Jesus come with the intent to “seize him” and they say, “He is out of his mind.” This raises the possibility that perhaps Mary and the other relatives were concerned about Jesus’ welfare.
Ultimately, though, when it comes to answering your question, we can only speculate. The good news is that we don’t really need to know, because the main point of the story is not the fact that these family members came, but what Jesus said in response to their arrival: “Whoever does the will of God is my brother and sister and mother.” (Mark 3:35). With this statement, Jesus is making it clear that kinship with him is not some elite club, but rather a state that is attainable by everyone who lives the life that God desires for us.