Objection 1. It would seem that "Gift" is not a personal name. For every personal name imports a distinction in God. But the name of "Gift" does not import a distinction in God; for Augustine says (De Trin. xv, 19): that "the Holy Ghost is so given as God's Gift, that He also gives Himself as God." Therefore "Gift" is not a personal name.
Objection 2. Further, no personal name belongs to the divine essence. But the divine essence is the Gift which the Father gives to the Son, as Hilary says (De Trin. ix). Therefore "Gift" is not a personal name.
Objection 3. Further, according to Damascene (De Fide Orth. iv, 19) there is no subjection nor service in the divine persons. But gift implies a subjection both as regards him to whom it is given, and as regards him by whom it is given. Therefore "Gift" is not a personal name.
Objection 4. Further, "Gift" imports relation to the creature, and it thus seems to be said of God in time. But personal names are said of God from eternity; as "Father," and "Son." Therefore "Gift" is not a personal name.
On the contrary, Augustine says (De Trin. xv, 19): "As the body of flesh is nothing but flesh; so the gift of the Holy Ghost is nothing but the Holy Ghost." But the Holy Ghost is a personal name; so also therefore is "Gift."
I answer that, The word "gift" imports an aptitude for being given. And what is given has an aptitude or relation both to the giver and to that to which it is given. For it would not be given by anyone, unless it was his to give; and it is given to someone to be his. Now a divine person is said to belong to another, either by origin, as the Son belongs to the Father; or as possessed by another. But we are said to possess what we can freely use or enjoy as we please: and in this way a divine person cannot be possessed, except by a rational creature united to God. Other creatures can be moved by a divine person, not, however, in such a way as to be able to enjoy the divine person, and to use the effect thereof. The rational creature does sometimes attain thereto; as when it is made partaker of the divine Word and of the Love proceeding, so as freely to know God truly and to love God rightly. Hence the rational creature alone can possess the divine person. Nevertheless in order that it may possess Him in this manner, its own power avails nothing: hence this must be given it from above; for that is said to be given to us which we have from another source. Thus a divine person can "be given," and can be a "gift."
Reply to Objection 1. The name "Gift" imports a personal distinction, in so far as gift imports something belonging to another through its origin. Nevertheless, the Holy Ghost gives Himself, inasmuch as He is His own, and can use or rather enjoy Himself; as also a free man belongs to himself. And as Augustine says (In Joan. Tract. xxix): "What is more yours than yourself?" Or we might say, and more fittingly, that a gift must belong in a way to the giver. But the phrase, "this is this one's," can be understood in several senses. In one way it means identity, as Augustine says (In Joan. Tract. xxix); and in that sense "gift" is the same as "the giver," but not the same as the one to whom it is given. The Holy Ghost gives Himself in that sense. In another sense, a thing is another's as a possession, or as a slave; and in that sense gift is essentially distinct from the giver; and the gift of God so taken is a created thing. In a third sense "this is this one's" through its origin only; and in this sense the Son is the Father's; and the Holy Ghost belongs to both. Therefore, so far as gift in this way signifies the possession of the giver, it is personally distinguished from the giver, and is a personal name.
Reply to Objection 2. The divine essence is the Father's gift in the first sense, as being the Father's by way of identity.
Reply to Objection 3. Gift as a personal name in God does not imply subjection, but only origin, as regards the giver; but as regards the one to whom it is given, it implies a free use, or enjoyment, as above explained.
Reply to Objection 4. Gift is not so called from being actually given, but from its aptitude to be given. Hence the divine person is called Gift from eternity, although He is given in time. Nor does it follow that it is an essential name because it imports relation to the creature; but that it includes something essential in its meaning; as the essence is included in the idea of person, as stated above (Question 34, Article 3).
Source: Summa Theologica (New Advent)