In the Summa Theologiae, St. Thomas assigns three proper names to the Holy Spirit: Holy Spirit ("Spiritus Sanctus"), Love ("Amor"), and Gift ("Donum"). (Similarly, the Son is properly named Son, Word, and Image.)
That the Holy Spirit is called the Holy Spirit is something all Christians should know. Most of us, I bet, could even fumble our way toward a reason. "Well, He's holy, right? And He's a spirit. So, yeah."
...both the Father is a Spirit, and the Son is a Spirit; and both the Father is holy, and the Son is holy -- as piety doubts not. And yet [the Holy Spirit] is specially called "the Holy Spirit"; for because He is common to both, He is specially called that which both are in common.
That the Holy Spirit is specially called "Love" may also sound familiar to a lot of Christians, who are used to saying things like, "The Holy Spirit is the love the Father and the Son have for each other." This too is thoroughly Augustinian:
And the Holy Spirit, according to the Holy Scriptures, is neither of the Father alone, nor of the Son alone, but of both; and so intimates to us a mutual love, wherewith the Father and the Son reciprocally love one another.
I suspect, though, that polling Christians on the question, "Who is the Gift of God?," would show a sizable response of, "Jesus." And why not? Isaiah, after all, famously prophesied, "A son is given to us."
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.
That is, both the Son and the Holy Spirit proceed from the Father. But the Son proceeds as Word, and to proceed as word is to be an image. The Holy Spirit, on the other hand, proceeds as Love, and to proceed as love is to be a gift. How so? According to St. Thomas,
a gift is properly ... 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, He proceeds as the first gift.
So when we speak of the "gifts of the Holy Spirit," we are presuming the first gift of the Spirit Himself.