Objection 1. It would seem that this name "Father" is not properly the name of a divine person. For the name "Father" signifies relation. Moreover "person" is an individual substance. Therefore this name "Father" is not properly a name signifying a Person.
Objection 2. Further, a begetter is more common than father; for every father begets; but it is not so conversely. But a more common term is more properly applied to God, as stated above (Question 13, Article 11). Therefore the more proper name of the divine person is begetter and genitor than Father.
Objection 3. Further, a metaphorical term cannot be the proper name of anyone. But the word is by us metaphorically called begotten, or offspring; and consequently, he of whom is the word, is metaphorically called father. Therefore the principle of the Word in God is not properly called Father.
Objection 4. Further, everything which is said properly of God, is said of God first before creatures. But generation appears to apply to creatures before God; because generation seems to be truer when the one who proceeds is distinct from the one whence it proceeds, not only by relation but also by essence. Therefore the name "Father" taken from generation does not seem to be the proper name of any divine person.
On the contrary, It is said (Psalm 88:27): "He shall cry out to me: Thou art my Father."
I answer that, The proper name of any person signifies that whereby the person is distinguished from all other persons. For as body and soul belong to the nature of man, so to the concept of this particular man belong this particular soul and this particular body; and by these is this particular man distinguished from all other men. Now it is paternity which distinguishes the person of the Father from all other persons. Hence this name "Father," whereby paternity is signified, is the proper name of the person of the Father.
Reply to Objection 1. Among us relation is not a subsisting person. So this name "father" among us does not signify a person, but the relation of a person. In God, however, it is not so, as some wrongly thought; for in God the relation signified by the name "Father" is a subsisting person. Hence, as above explained (29, 4), this name "person" in God signifies a relation subsisting in the divine nature.
Reply to Objection 2. According to the Philosopher (De Anima ii, text 49), a thing is denominated chiefly by its perfection, and by its end. Now generation signifies something in process of being made, whereas paternity signifies the complement of generation; and therefore the name "Father" is more expressive as regards the divine person than genitor or begettor.
Reply to Objection 3. In human nature the word is not a subsistence, and hence is not properly called begotten or son. But the divine Word is something subsistent in the divine nature; and hence He is properly and not metaphorically called Son, and His principle is called Father.
Reply to Objection 4. The terms "generation" and "paternity" like the other terms properly applied to God, are said of God before creatures as regards the thing signified, but not as regards the mode of signification. Hence also the Apostle says, "I bend my knee to the Father of my Lord Jesus Christ, from whom all paternity in heaven and on earth is named" (Ephesians 3:14). This is explained thus. It is manifest that generation receives its species from the term which is the form of the thing generated; and the nearer it is to the form of the generator, the truer and more perfect is the generation; as univocal generation is more perfect than non-univocal, for it belongs to the essence of a generator to generate what is like itself in form. Hence the very fact that in the divine generation the form of the Begetter and Begotten is numerically the same, whereas in creatures it is not numerically, but only specifically, the same, shows that generation, and consequently paternity, is applied to God before creatures. Hence the very fact that in God a distinction exists of the Begotten from the Begetter as regards relation only, belongs to the truth of the divine generation and paternity.
Source: Summa Theologica (New Advent)