The Lord intended that He would leave His grieving mother in the best of care. It is not certain that at that time His brothers believed in Him.
John 7:3 So his brothers said to him, “Leave here and go to Judea, that your disciples also may see the works you are doing. 4 For no one works in secret if he seeks to be known openly. If you do these things, show yourself to the world.” 5 For not even his brothers believed in him.
If we remember that the whole Gospel of John takes a very short "snapshot" of the life of Jesus (~22 days) then we can see why The Lord left the care of His mother to John.
John was a believer, His brothers were not at this time, and since His mother was a believer, it was better that she be taken care of by another believer.
I deduce from this that it is more important to be in fellowship with God's people than all the world, even our own siblings/family.