Objection 1. It would seem that Gift is not the proper name of the Holy Ghost. For the name Gift comes from being given. But, as Isaiah 9:16 says: "A Son is give to us." Therefore to be Gift belongs to the Son, as well as to the Holy Ghost.
Objection 2. Further, every proper name of a person signifies a property. But this word Gift does not signify a property of the Holy Ghost. Therefore Gift is not a proper name of the Holy Ghost.
Objection 3. Further, the Holy Ghost can be called the spirit of a man, whereas He cannot be called the gift of any man, but "God's Gift" only. Therefore Gift is not the proper name of the Holy Ghost.
On the contrary, Augustine says (De Trin. iv, 20): "As 'to be born' is, for the Son, to be from the Father, so, for the Holy Ghost, 'to be the Gift of God' is to proceed from Father and Son." But the Holy Ghost receives His proper name from the fact that He proceeds from Father and Son. Therefore Gift is the proper name of the Holy Ghost.
I answer that, Gift, taken personally in God, is the proper name of the Holy Ghost.
In proof of this we must know that a gift is properly an unreturnable giving, as Aristotle says (Topic. iv, 4)--i.e. a thing which is not given with the intention of a return--and it thus contains the idea of a gratuitous donation. Now, the reason of donation being gratuitous is love; since therefore do we give something to anyone gratuitously forasmuch as we wish him well. So what we first give him is the love whereby we wish him well. Hence it is manifest that love has the nature of a first gift, through which all free gifts are given. So since the Holy Ghost proceeds as love, as stated above (27, 4; 37, 1), He proceeds as the first gift. Hence Augustine says (De Trin. xv, 24): "By the gift, which is the Holy Ghost, many particular gifts are portioned out to the members of Christ."
Reply to Objection 1. As the Son is properly called the Image because He proceeds by way of a word, whose nature it is to be the similitude of its principle, although the Holy Ghost also is like to the Father; so also, because the Holy Ghost proceeds from the Father as love, He is properly called Gift, although the Son, too, is given. For that the Son is given is from the Father's love, according to the words, "God so loved the world, as to give His only begotten Son" (John 3:16).
Reply to Objection 2. The name Gift involves the idea of belonging to the Giver through its origin; and thus it imports the property of the origin of the Holy Ghost--that is, His procession.
Reply to Objection 3. Before a gift is given, it belongs only to the giver; but when it is given, it is his to whom it is given. Therefore, because "Gift" does not import the actual giving, it cannot be called a gift of man, but the Gift of God giving. When, however, it has been given, then it is the spirit of man, or a gift bestowed on man.
Source: Summa Theologica (New Advent)