Yes. The Holy Ghost is a personal name and so also therefore is "Gift" because that is said to be given to us which we have from another source.
In one 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.
The divine essence is the Father's gift in the first sense, as being the Father's by way of identity.
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 (Q34, A3).