Since Jesus had brothers and sisters (Mark 6:3), why would the disciple John be given apparent responsibility for Mary, the mother of Jesus in John 19:25-27?† Wouldnít it seem better for Mary to be cared for by her own children?† This would be particularly so because two of them, James and Jude, apparently wrote books of the Bible. The authors of the epistles of James and Jude are both reported to have been half brothers of Jesus Christ.†
The simple answer is because Jesus' brothers and sisters were not there when Jesus was crucified. While Jesus was alive, his siblings did not believe he was the Christ. "For even His brothers did not believe in Him" (John 7:5).
At the end of his life, Jesus wanted his mother cared for by someone he could trust, so he placed her care in the hands of the disciple whom he loved, who also was present at his death. "Now there stood by the cross of Jesus His mother, and His mother's sister, Mary the wife of Clopas, and Mary Magdalene. When Jesus therefore saw His mother, and the disciple whom He loved standing by, He said to His mother, "Woman, behold your son!" Then He said to the disciple, "Behold your mother!" And from that hour that disciple took her to his own home" (John 19:25-27). As the eldest son in the family, Jesus fulfilled his duty to his parent.
I can imagine that Jesus' brothers felt slighted by his choice. But it remains a fact that Mary believed in Jesus and they did not. Jesus' choice was both logical and practical.
It is true that later, one of Jesus' brothers, James, did become a Christian and a leading member in the church (Galatians 1:19). But this doesn't change what took place at the cross.
Neither author of James or Jude claims to be the Lord's brother. The author of Jude does mention that he has a brother named James. The problem is that there are a number of James and Judes mentioned in the Bible and there are two sets of possible brothers who are likely to have been the authors of the two books. See: "Can you help me understand who were the people with similar names?"
In my humble opinion thatís a great answer. Thanks!