Objection 1. It would seem that this term "person" cannot be common to the three persons. For nothing is common to the three persons but the essence. But this term "person" does not signify the essence directly. Therefore it is not common to all three.
Objection 2. Further, the common is the opposite to the incommunicable. But the very meaning of person is that it is incommunicable; as appears from the definition given by Richard of St. Victor (29, 3, ad 4). Therefore this term "person" is not common to all the three persons.
Objection 3. Further, if the name "person" is common to the three, it is common either really, or logically. But it is not so really; otherwise the three persons would be one person; nor again is it so logically; otherwise person would be a universal. But in God there is neither universal nor particular; neither genus nor species, as we proved above (Question 3, Article 5). Therefore this term "person" is not common to the three.
On the contrary, Augustine says (De Trin. vii, 4) that when we ask, "Three what?" we say, "Three persons," because what a person is, is common to them.
I answer that, The very mode of expression itself shows that this term "person" is common to the three when we say "three persons"; for when we say "three men" we show that "man" is common to the three. Now it is clear that this is not community of a real thing, as if one essence were common to the three; otherwise there would be only one person of the three, as also one essence.
What is meant by such a community has been variously determined by those who have examined the subject. Some have called it a community of exclusion, forasmuch as the definition of "person" contains the word "incommunicable." Others thought it to be a community of intention, as the definition of person contains the word "individual"; as we say that to be a "species" is common to horse and ox. Both of these explanations, however, are excluded by the fact that "person" is not a name of exclusion nor of intention, but the name of a reality. We must therefore resolve that even in human affairs this name "person" is common by a community of idea, not as genus or species, but as a vague individual thing. The names of genera and species, as man or animal, are given to signify the common natures themselves, but not the intentions of those common natures, signified by the terms "genus" or "species." The vague individual thing, as "some man," signifies the common nature with the determinate mode of existence of singular things--that is, something self-subsisting, as distinct from others. But the name of a designated singular thing signifies that which distinguishes the determinate thing; as the name Socrates signifies this flesh and this bone. But there is this difference--that the term "some man" signifies the nature, or the individual on the part of its nature, with the mode of existence of singular things; while this name "person" is not given to signify the individual on the part of the nature, but the subsistent reality in that nature. Now this is common in idea to the divine persons, that each of them subsists distinctly from the others in the divine nature. Thus this name "person" is common in idea to the three divine persons.
Reply to Objection 1. This argument is founded on a real community.
Reply to Objection 2. Although person is incommunicable, yet the mode itself of incommunicable existence can be common to many.
Reply to Objection 3. Although this community is logical and not real, yet it does not follow that in God there is universal or particular, or genus, or species; both because neither in human affairs is the community of person the same as community of genus or species; and because the divine persons have one being; whereas genus and species and every other universal are predicated of many which differ in being.
Source: Summa Theologica (New Advent)