Objection 1. It would seem that this name, "Holy Ghost," is not the proper name of one divine person. For no name which is common to the three persons is the proper name of any one person. But this name of 'Holy Ghost' [It should be borne in mind that the word "ghost" is the old English equivalent for the Latin "spiritus," whether in the sense of "breath" or "blast," or in the sense of "spirit," as an immaterial substance. Thus, we read in the former sense (Hampole, Psalter x, 7), "The Gost of Storms" [spiritus procellarum], and in the latter "Trubled gost is sacrifice of God" (Prose Psalter, A.D. 1325), and "Oure wrestlynge is... against the spiritual wicked gostes of the ayre" (More, "Comfort against Tribulation"); and in our modern expression of "giving up the ghost." As applied to God, and not specially to the third Holy Person, we have an example from Maunder, "Jhesu Criste was the worde and the goste of Good." (See Oxford Dictionary).] is common to the three persons; for Hilary (De Trin. viii) shows that the "Spirit of God" sometimes means the Father, as in the words of Isaiah 61:1: "The Spirit of the Lord is upon me;" and sometimes the Son, as when the Son says: "In the Spirit of God I cast out devils" (Matthew 12:28), showing that He cast out devils by His own natural power; and that sometimes it means the Holy Ghost, as in the words of Joel 2:28: "I will pour out of My Spirit over all flesh." Therefore this name 'Holy Ghost' is not the proper name of a divine person.
Objection 2. Further, the names of the divine persons are relative terms, as Boethius says (De Trin.). But this name "Holy Ghost" is not a relative term. Therefore this name is not the proper name of a divine Person.
Objection 3. Further, because the Son is the name of a divine Person He cannot be called the Son of this or of that. But the spirit is spoken of as of this or that man, as appears in the words, "The Lord said to Moses, I will take of thy spirit and will give to them" (Numbers 11:17) and also "The Spirit of Elias rested upon Eliseus" (2 Kings 2:15). Therefore "Holy Ghost" does not seem to be the proper name of a divine Person.
On the contrary, It is said (1 John 5:7): "There are three who bear witness in heaven, the Father, the Word, and the Holy Ghost." As Augustine says (De Trin. vii, 4): "When we ask, Three what? we say, Three persons." Therefore the Holy Ghost is the name of a divine person.
I answer that, While there are two processions in God, one of these, the procession of love, has no proper name of its own, as stated above (27, 4, ad 3). Hence the relations also which follow from this procession are without a name (28, 4): for which reason the Person proceeding in that manner has not a proper name. But as some names are accommodated by the usual mode of speaking to signify the aforesaid relations, as when we use the names of procession and spiration, which in the strict sense more fittingly signify the notional acts than the relations; so to signify the divine Person, Who proceeds by way of love, this name "Holy Ghost" is by the use of scriptural speech accommodated to Him. The appropriateness of this name may be shown in two ways.
Firstly, from the fact that the person who is called "Holy Ghost" has something in common with the other Persons. For, as Augustine says (De Trin. xv, 17; v, 11), "Because the Holy Ghost is common to both, He Himself is called that properly which both are called in common. For the Father also is a spirit, and the Son is a spirit; and the Father is holy, and the Son is holy."
Secondly, from the proper signification of the name. For the name spirit in things corporeal seems to signify impulse and motion; for we call the breath and the wind by the term spirit. Now it is a property of love to move and impel the will of the lover towards the object loved. Further, holiness is attributed to whatever is ordered to God. Therefore because the divine person proceeds by way of the love whereby God is loved, that person is most properly named "The Holy Ghost."
Reply to Objection 1. The expression Holy Spirit, if taken as two words, is applicable to the whole Trinity: because by 'spirit' the immateriality of the divine substance is signified; for corporeal spirit is invisible, and has but little matter; hence we apply this term to all immaterial and invisible substances. And by adding the word "holy" we signify the purity of divine goodness. But if Holy Spirit be taken as one word, it is thus that the expression, in the usage of the Church, is accommodated to signify one of the three persons, the one who proceeds by way of love, for the reason above explained.
Reply to Objection 2. Although this name "Holy Ghost" does not indicate a relation, still it takes the place of a relative term, inasmuch as it is accommodated to signify a Person distinct from the others by relation only. Yet this name may be understood as including a relation, if we understand the Holy Spirit as being breathed [spiratus].
Reply to Objection 3. In the name Son we understand that relation only which is of something from a principle, in regard to that principle: but in the name "Father" we understand the relation of principle; and likewise in the name of Spirit inasmuch as it implies a moving power. But to no creature does it belong to be a principle as regards a divine person; but rather the reverse. Therefore we can say "our Father," and "our Spirit"; but we cannot say "our Son."
Source: Summa Theologica (New Advent)