Well, we aren’t really 100% sure which of the two readings is the original (i.e. the one Luke actually wrote); you really have two possibilities here:
(1) Luke really wrote “I am well-pleased” (in agreement with Matthew and Mark) and the “begotten” reading is the secondary one. In other words, some copyist/s turned God’s saying into a more obvious quotation from Psalm 2.
(2) Luke really wrote “I have begotten you,” but later copyists substituted “well-pleased” instead, either for the sake of harmonization (to conform Luke with the other two synoptics) or force of habit (since “well-pleased” is the reading found in Matthew and Mark anyway) or because the “begotten” reading was felt to have the potential of being misused for Adoptionism.
You shouldn’t be surprised here; both are something that are known to happen in the transmission process from time to time. That’s why there are manuscripts which say Jesus “moved with compassion” on the leper who came to Him, and a few manuscripts which say that He was “moved with anger” instead. That’s also why later manuscripts of Luke give Matthew’s version of the Lord’s prayer (“Our Father who art in heaven …”) instead of the more concise version (“Father, hallowed be your name …”) found in earlier manuscripts.
St. Augustine was one of the ancient authors who we know know the variant. He observes that by his time at least the reading “is said not to be found in the more ancient Greek codices, yet if it can be established by any copies worthy of credit, what results but that we suppose both voices to have been heard from heaven, in one or other verbal order?” In other words, in his idea, maybe God said both.
I should add: the word for “begotten” here is the verb
gegennēka - “I have fathered you.”
Monogenēs (which is an adjective BTW) is traditionally translated as “only-begotten,” but it’s more likely to mean “only one,” “unique.”
