According to Catholic theology, why does God have an only Son?

God's plans are centred around the father glorifying the son and giving him a kingdom and a bride. We can see this in a few passages from the NT:

In the past God spoke to our ancestors through the prophets at many times and in various ways, but in these last days he has spoken to us by his Son, whom he appointed heir of all things, and through whom also he made the universe. The Son is the radiance of God’s glory and the exact representation of his being, sustaining all things by his powerful word. After he had provided purification for sins, he sat down at the right hand of the Majesty in heaven. So he became as much superior to the angels as the name he has inherited is superior to theirs. (Hebrews 1:1-4, NIV)

But about the Son he says,
 “Your throne, O God, will last for ever and ever;
  a scepter of justice will be the scepter of your kingdom.
 You have loved righteousness and hated wickedness;
  therefore God, your God, has set you above your companions
  by anointing you with the oil of joy.” (Hebrews 1:8-9, NIV)

... That power is the same as the mighty strength he exerted when he raised Christ from the dead and seated him at his right hand in the heavenly realms, far above all rule and authority, power and dominion, and every name that is invoked, not only in the present age but also in the one to come. And God placed all things under his feet and appointed him to be head over everything for the church, which is his body, the fullness of him who fills everything in every way. (Ephesians 1:19-23, NIV)

Husbands, love your wives, just as Christ loved the church and gave himself up for her to make her holy, cleansing her by the washing with water through the word, and to present her to himself as a radiant church, without stain or wrinkle or any other blemish, but holy and blameless. ... This is a profound mystery—but I am talking about Christ and the church. (Ephesians 5:25-27, 32, NIV)

Trinitarians believe that the differences between the persons of the trinity are differences of roles, not of substance, except that son became a human at the incarnation. There is only one son of God because of what that role means. There is only one son because there is only one

• heir of the universe
• seat at the father's right hand
• throne for the authority appointed over everyone else
• redeemer of the human race
• groom for the church

While it seems theoretically possible for the godhead to have more than three persons, we have to admit that it's ultimately beyond our understanding, and we have to make the best of the little God has decided to reveal about himself in the scriptures. Theoretically there could have been two persons of the trinity who were the children of God, but God's plans to glorify the son mean there could only be one person called "the son".

Reformed protestants believe that this was the plan since before the creation of the universe, an idea called the "covenant of redemption". You may be interested to read my question asking for the biblical basis for this belief.

The Catechism of the Catholic Church contains a section on the line of the Apostles' Creed reading (in translation) "and in Jesus Christ, His only Son, our Lord". Each phrase is discussed separately; but unfortunately the five paragraphs discussing the phrase "His only Son" focus more on "Son" than on "only".

St. Thomas Aquinas, however, touches on the issue in the Summa Theologica. The First Part of this work contains a large number of "questions" (the Latin might be better translated "discussions" or "investigations" regarding the Trinity. In particular, Question 27, Article 5 (and again the Latin might be better translated "point" or "section") asks "Whether there are more than two processions in God?"

Aquinas explains the relationship of the Father to the Son by offering the analogy of a person who has in mind something they want to say. In one sense, the word they want to say is distinct from them; they are not identical to what they say. In another sense, though, the word (especially before it is spoken) is a part of them: it is a thought in their mind. Thus, Aquinas says, the Son is to the Father, proceeding from Him as one's word proceeds from one's mind and yet part of Him as one's words and thoughts are part of who one is. (In a similar way, the Spirit proceeds from the Father and the Son as a person's love proceeds from them and from their words.)

Aquinas derives from this analogy an objection to God's having only one Son: "In us there is not only one procession of the word, but there are many: for in us from one word proceeds another. ... Therefore in God there are more than two processions." (Question 27, Article 5, Objection 3) That is, he objects that since we can have multiple words proceeding from us, surely God could have multiple Words (that is, multiple Sons of God) proceeding from Him?

His response is to say that God is not really like us in this sense. Words, in us, proceed from our understanding of a situation; and our need to use multiple words arises from the fact that our understanding comes in parts and stages. We don't automatically understand everything perfectly all at once, nor can we (therefore) communicate everything perfectly all at once. But God can. He has therefore, Aquinas says, only one Word by which he understands everything, and communicates everything: "God understands all things by one simple act. ... Hence there cannot exist in Him a procession of Word from Word ... for there is in Him only one perfect Word." (Reply to Objection 3)

This unique Word of God we call the Son of God (as Aquinas discusses in Question 27 Article 2); and thus there can be only one Son of God.