Objection 1. It would seem that it was not more fitting that the Son of God should become incarnate than the Father or the Holy Ghost. For by the mystery of Incarnation men are led to the true knowledge of God, according to John 18:37: "For this was I born, and for this came I into the world, to give testimony to the truth." But by the Person of the Son of God becoming incarnate many have been kept back from the true knowledge of God, since they referred to the very Person of the Son what was said of the Son in His human nature, as Arius, who held an inequality of Persons, according to what is said (John 14:28): "The Father is greater than I." Now this error would not have arisen if the Person of the Father had become incarnate, for no one would have taken the Father to be less than the Son. Hence it seems fitting that the Person of the Father, rather than the Person of the Son, should have become incarnate.
Objection 2. Further, the effect of Incarnation would seem to be, as it were, a second creation of human nature, according to Galatians 6:15: "For in Christ Jesus neither circumcision availeth anything, nor uncircumcision, but a new creature." But the power of creation is appropriated to the Father. Therefore it would have been more becoming to the Father than to the Son to become incarnate.
Objection 3. Further, Incarnation is ordained to the remission of sins, according to Matthew 1:21: "Thou shalt call His name Jesus. For He shall save His people from their sins." Now the remission of sins is attributed to the Holy Ghost according to John 20:22-23: "Receive ye the Holy Ghost. Whose sins you shall forgive, they are forgiven them." Therefore it became the Person of the Holy Ghost rather than the Person of the Son to become incarnate.
On the contrary, Damascene says (De Fide Orth. iii, 1): "In the mystery of Incarnation the wisdom and power of God are made known: the wisdom, for He found a most suitable discharge for a most heavy debt; the power, for He made the conquered conquer." But power and wisdom are appropriated to the Son, according to 1 Corinthians 1:24: "Christ, the power of God and the wisdom of God." Therefore it was fitting that the Person of the Son should become incarnate.
I answer that, 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 (Sirach 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 Romans 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 (Romans 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 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.
Reply to Objection 1. There is nothing which human malice cannot abuse, since it even abuses God's goodness, according to Romans 2:4: "Or despisest thou the riches of His goodness?" Hence, even if the Person of the Father had become incarnate, men would have been capable of finding an occasion of error, as though the Son were not able to restore human nature.
Reply to Objection 2. The first creation of things was made by the power of God the Father through the Word; hence the second creation ought to have been brought about through the Word, by the power of God the Father, in order that restoration should correspond to creation according to 2 Corinthians 5:19: "For God indeed was in Christ reconciling the world to Himself."
Reply to Objection 3. To be the gift of the Father and the Son is proper to the Holy Ghost. But the remission of sins is caused by the Holy Ghost, as by the gift of God. And hence it was more fitting to man's justification that the Son should become incarnate, Whose gift the Holy Ghost is.
Source: Summa Theologica (New Advent)