Thank you, my dear friend, for such an interesting question!  The verses you refer to are found in John 19:26-27. I do not think that I can point to specific Bible verses that will tell us precisely why the Lord chose John and no other, but I think I can point us to some Scriptures that will help to gain some insight into the Lord’s probable purpose in this decision. I believe that the answer to your question is most likely that the Lord Jesus chose one for this honor whom He knew was a true believer on Him, and one who would be around (alive) to be able to take care of Mary until her natural death. I know that while the Lord Jesus was true God, He was also true man; and as a man of Israel, He would have the responsibility of seeing to the care of dependent relatives such as a widowed mother in the event of the son’s death. Thus, while on the cross, dying for lost sinners like you and me, the Lord Jesus did take care of this responsibility for Mary who was His earthly mother. His mother was certainly a widow by this time and so she would need to be provided for, He seeing to this as her eldest Son. Typically, this responsibility might have gone to one of His half-brothers, but I do not believe that any of these were believers by this time. So, the Lord gave this responsibility to John, one of His very closest friends and a faithful disciple, and one who would live in Jerusalem for quite some time. 

It should be noted here that we do not read of any of the Lord’s half-brothers being believers on Christ Jesus until after He ascended into heaven. We read in John 7:5, “For neither did His brethren believe in Him.” I truly believe that the Lord Jesus would not leave His earthly mother to be cared for by an unbeliever, and I do not think any of his half-brothers came to believe on Him right away.  Now, we do know that James, the half-brother of the Lord Jesus, did ultimately come to be a Christian, and we know that he later would author the book of James in the Bible. We do read that the Lord appeared to James in 1 Corinthians 15:7, “After that, He was seen of James; then of all the apostles.” Most commentators feel that James in this verse is the Lord’s half-brother. I thus leap to the conclusion that upon seeing the Lord Jesus alive, James finally did believe, but not until after Christ had risen. Another verse possibly supporting the idea that the half-brothers did not become Christians until the time of His ascension into heaven might be Acts 1:14 where we read, ”These all continued with one accord in prayer and supplication, with the women, and Mary the mother of Jesus, and with His brethren.” Regarding this last verse, MacDonald says: “The mention of the brothers of Jesus, coming after the reference to Mary, makes it likely that these were actual sons of Mary and half-brothers of Jesus.” But again, we do not hear of them gathering together as believers until after the ascension.

To conclude, I do not know that we are told directly by the Scriptures why it was that the Lord Jesus chose John to look after His earthly mother after His death on the cross. We do see that the Lord Jesus, as He was about to give up the ghost, directed John, a faithful disciple whom the Lord Jesus was close to in this scene to take Mary into his own home after His death.  Why it was John and not one of the other of His disciples we are not told, but I suspect it was because each of the apostles would be martyred except John.

I hope this is helpful to you in looking at this question.  As a side note, though, I believe that it is important to bring out that the Scriptures do not elevate or attribute any special role or title to Mary His earthly mother, nor to His half-brothers. We do not read further of Mary after Acts 1, and neither James nor his brothers were designated apostles.  I bring this out because all of this is to verify that Jesus, while being born into this world as the son of Mary, was actually the Son of God, who came into the world to save sinners.  Neither Mary nor the half-brothers of Jesus had a special place above other human beings in this world, and we do not see the Scriptures giving them any special rank or place. True it is that Mary was highly honored to be the earthly mother of the Christ of God, but she was after all a sinner, like you and me. Likewise, just because Jesus had some half-brothers according to His humanity, these too were sinners in need of a Savior. All in this world, and this is true of Mary and her other children, must come on the same ground as any other sinner.  We are all saved by grace through faith, upon repentance and trust in the Lord as our Savior. And who indeed are those who are closest to the Lord Jesus? The Lord Jesus did say on one occasion, when His earthly family came to call after Him” For whosoever shall do the will of God, the same is My brother, and my sister, and mother,” (Mark 3:35). Thus, I believe it makes sense that the Lord Jesus should arrange the care of His earthly mother with someone of the household of faith, and one who would live long enough to take care of this responsibility throughout Mary’s remaining life.  (SF)  (591.2)