Tony Mangini recently asked a great question in the comments:
I wonder why Jesus on at least three occasions brought Peter, James and John with him but left the other apostles out. Why, in addition to the fact that Jesus loved John, do you think he used only three?
This great questions gives us the opportunity to reflect on how Old Testament Judaism is the origin of New Testament Catholicism.
In the Old Covenant, Moses had seventy(two) elders. Each of the twelve tribes had a chief. Finally, Moses had three close men who went up on the mountain with him: Aaron, Nadab, and Abiu:
“AND he said to Moses: Come up to the Lord, thou, and Aaron, Nadab and Abiu, and seventy of the elders of Israel, and you shall adore afar off.” (Exodus 24:1, D-R)
In the New Covenant, Christ is re-establishing Israel. The Catholic Church is the new Israel. Christ appoint seventy(two) elders in Luke 10:1-24. He also appoints twelve Apostles. Then, like Moses, he has three "inside men" who form His inner circle. Just as Aaron was the High Priest for Moses, so Peter serves as the High Priest or Pope.
James and John are special for three reasons:
1. First, James is the first Apostle to die and John is the last Apostle to die. They are the temporal bookends of the Apostolic ministry.
2. Second, James is the first martyr of the Apostles. John, is the only non-martyr of the Apostles.
3. Third, James was the first to receive an apparition of the Blessed Virgin and John was given the Virgin at the crucifixion and served as her guardian till her holy Dormition.
Essentially, these three apostles were more intimately associated with the suffering of Christ: first Pope (Peter), first Martyr (James), only Apostle at the crucifixion and guardian of Mary (John).
Their intense suffering required them to be present at the Transfiguration and the Agony in the Garden.