From St. Augustine's Two Books on Genesis Against the Manichees, bk. 2, pp. 111-12:
Man's Work in Paradise and Woman Is
Made As His Helper
Although man was placed in paradise so as to work and guard it, [Gen. 2:15] that praiseworthy work was not toilsome. For the work in paradise is quite different from the work on the earth to which he was condemned after the sin. The addition "and to guard it" indicated the sort of work it was. For in the tranquility of the happy life, where there is no death, the only work is to guard what you possess. He also receives the command that we have already dealt with above. This command ends so that it is not addressing one person, for it says, "On the day that you eat, you shall die the death." [Gen. 2:17] Hence, Scripture begins to explain how the woman was made. It says that she was made as man's helper so that by spiritual union she might bring forth spiritual offspring, that is, the good works of divine praise, while he rules and she obeys. He is ruled by wisdom, she by the man. For Christ is the head of the man, and the man is the head of the woman. [1 Cor. 11:3] Thus it said, "It is not good that man is alone." [Gen. 2:18] For there was still need to bring it about not only that the soul rule over the body, because the body has the position of a servant, but also that virile reason hold subject to itself its animal part, by the help of which it governs the body. The woman was made as an illustration of this, for the order of things makes her subject to man. Thus we can also come to see in one human what we can see more clearly in two humans, that is, in the male and the female. The interior mind, like virile reason, should hold subject the soul's appetite by means of which we control the members of the body, and by just law it should place a limit upon its helper, just as man ought to rule woman and ought not to allow her to rule him. When this happens, the home is perverted and unhappy.
Hence, God first showed man how much better he was than the cattle and all irrational animals, and this is signified by the statement that all the animals were brought to him that he might see what he would call them and give them names. [Gen. 2:19-20] This shows that man is better than the animals in virtue of reason, since only reason which judges concerning them is able to distinguish and know them by name. This latter idea is an easy one to grasp, for man quickly understands that he is better than the cattle. The former idea is a difficult one to grasp, namely, that by which he understands that the rational part in him that rules is distinct from the animal part which is ruled.
Thus, according to St. Augustine, Adam named the animals to show that he is distinct from and superior to them (having an intellect, unlike them) and that he is to rule them, just as his soul is distinct from and superior to his body, which must be subjected to reason. Similarly, man must rule woman and not let woman overrule him.