Is there an explanation in Catholicism as to why it was the Son and not the others of the Blessed Trinity to become man and be crucified for our sake?

Question

According to Catholic faith and theology, once God decreed to save mankind, redemption would be by the work of God-made-man.

For us men and for our salvation,
he came down from heaven: [...]
and became man.
For our sake he was crucified. - The Nicene Creed.

And

But God decided that man should be shown mercy and be pardoned. His mercy was to be above all His works; and He also decreed that this mercy and pardon were to be granted because of the full satisfaction of justice by adequate atonement. Hence it became necessary that One of the Trinity assume created nature and in that nature offer reparation to God for man's great sin. - Library: The theology of the Precious Blood | Catholic Culture.

Within Catholicism is there an explanation of why it was the Son and not the Father or the Holy Spirit who for us men and our salvation became man and was crucified?

Answer 1

St. Thomas Aquinas addresses this question in Summa Theologica III q. 3 a. 8 ("Whether it was more fitting that the Person of the Son rather than any other Divine Person should assume human nature?") c.:

It was most fitting that the Person of the Son should become incarnate. First, on the part of the union; for such as are similar are fittingly united. Now the Person of the Son, Who is the Word of God, has a certain common agreement with all creatures, because the word of the craftsman, i.e. his concept, is an exemplar likeness of whatever is made by him. Hence the Word of God, Who is His eternal concept, is the exemplar likeness of all creatures. And therefore as creatures are established in their proper species, though movably, by the participation of this likeness, so by the non-participated and personal union of the Word with a creature, it was fitting that the creature should be restored in order to its eternal and unchangeable perfection; for the craftsman by the intelligible form of his art, whereby he fashioned his handiwork, restores it when it has fallen into ruin. Moreover, He has a particular agreement with human nature, since the Word is a concept of the eternal Wisdom, from Whom all man's wisdom is derived. And hence man is perfected in wisdom (which is his proper perfection, as he is rational) by participating the Word of God, as the disciple is instructed by receiving the word of his master. Hence it is said (Ecclus. 1:5): "The Word of God on high is the fountain of wisdom." And hence for the consummate perfection of man it was fitting that the very Word of God should be personally united to human nature.

Secondly, the reason of this fitness may be taken from the end of the union, which is the fulfilling of predestination, i.e. of such as are preordained to the heavenly inheritance, which is bestowed only on sons, according to Rm. 8:17: "If sons, heirs also." Hence it was fitting that by Him Who is the natural Son, men should share this likeness of sonship by adoption, as the Apostle says in the same chapter (Rm. 8:29): "For whom He foreknew, He also predestinated to be made conformable to the image of His Son."

Thirdly, the reason for this fitness may be taken from the sin of our first parent, for which the Incarnation supplied the remedy. For the first man sinned by seeking knowledge, as is plain from the words of the serpent, promising to man the knowledge of good and evil. Hence it was fitting that by the Word of true knowledge man might be led back to God, having wandered from God through an inordinate thirst for knowledge.

The objections to the question and St. Thomas's replies are interesting, too.

Answer 2

Short Answer

It could not be the Holy Spirit, as the Holy Spirit doesn't have a body (beyond the Son's, more on that later).

It could not be the Father, since God is not bound by death. (God is eternal: always is, was and shall be, as taught by the Catholic Church).

Who does that leave? The Son.

Discussion

The Doctrine of the Trinity holds that the Triune God is three persons sharing one essence/one substance (the Son being "consubstantial" with the Father is in the Nicene Creed) -- only one of whom had a mortal/fleshy nature to give up in sacrifice.
(CCC = Catechism of the Catholic Church)

CCC 198

198 Father is the first divine person of the Most Holy Trinity

CCC 430

430 Jesus means in Hebrew: "God saves." At the annunciation, the angel Gabriel gave him the name Jesus as his proper name, which expresses both his identity and his mission. Since God alone can forgive sins, it is God who, in Jesus his eternal Son made man, "will save his people from their sins". In Jesus, God recapitulates all of his history of salvation on behalf of men.

The Second Person is revealed in the flesh: why? (CCC 458-459)

458 The Word became flesh so that thus we might know God's love: "In this the love of God was made manifest among us, that God sent his only Son into the world, so that we might live through him."(1 John 4:9) "For God so loved the world that he gave his only Son, that whoever believes in him should not perish but have eternal life." (John 3:16)

459 The Word became flesh to be our model of holiness: "Take my yoke upon you, and learn from me." Mt 11:29; ⇒ "I am the way, and the truth, and the life; no one comes to the Father, but by me." Jn 14:6. "On the mountain of the Transfiguration, the Father commands: "Listen to him!" (Mark 9:7) Jesus is the model for the Beatitudes and the norm of the new law: "Love one another as I have loved you." (John 15:12) This love implies an effective offering of oneself, after his example.

CCC 684-685

684 Through his grace, the Holy Spirit is the first to awaken faith in us and to communicate to us the new life, which is to "know the Father and the one whom he has sent, Jesus Christ." But the Spirit is the last of the persons of the Holy Trinity to be revealed. St. Gregory of Nazianzus, the Theologian, explains this progression in terms of the pedagogy of divine "condescension":

The Old Testament proclaimed the Father clearly, but the Son more obscurely. The New Testament revealed the Son and gave us a glimpse of the divinity of the Spirit. Now the Spirit dwells among us and grants us a clearer vision of himself. It was not prudent, when the divinity of the Father had not yet been confessed, to proclaim the Son openly and, when the divinity of the Son was not yet admitted, to add the Holy Spirit as an extra burden, to speak somewhat daringly.... By advancing and progressing "from glory to glory," the light of the Trinity will shine in ever more brilliant rays.

685 To believe in the Holy Spirit is to profess that the Holy Spirit is one of the persons of the Holy Trinity, consubstantial with the Father and the Son: "with the Father and the Son he is worshipped and glorified." For this reason, the divine mystery of the Holy Spirit was already treated in the context of Trinitarian "theology."

Put another way: of the three persons in the Trinity, only one had the characteristic of being mortal/human that could giving up a life for any reason. (The Salvation of Mankind).

The other point is less canonical and more practical, and involves setting an example as a form of sending a message.

From First to Last ...

Beginning/Baptism: Jesus was already divine (being of both divine and human nature, per Catholic Dogma and most Christian belief) so why did he go into the River Jordan to let John the Baptist baptize him? Jesus, didn't need to repent from sin as those who were coming to John's baptisms need to. Jesus set by his actions the perfect example for how to turn away from sin and back towards God. (No further digression into Baptism, as that isn't the question).

Middle/Ministry: Jesus' ministry was about teaching by word and deed how to turn back towards God, and how to better fulfill the role of being God's children. About a thousand years later, St. Francis provided the following advice to all Christians in terms of how to be a better disciple: "Preach the Gospel every day. Use words when you have to."¹

Jesus didn't just talk the talk, he walked the walk. His ministry as recorded in the Gospels demonstrated (among other things) how to live correctly (as God had intended for us to).

End: Crucifixion/Sacrifice.

The crucifixion can be seen through the lens of Jesus' command to his disciples: "love one another as I have loved you." Love in this sense is best understood as agape -- which is a self-sacrificial love, a divine love, and a love that can hurt. Jesus set the extreme example of how much of a sacrifice one man can make for his fellow man by giving it all up, to include His life.

That example only works as a message (actions speaking louder than words) if you are made of flesh in the first place. If you are a being made wholly of spirit, you can't sacrifice a mortal life, and thus can't set this example.

The message obviously got across. (Pun intended).

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¹ The provenance of that aphorism has been challenged, but it is consistent with St. Francis' message.