How is the Son begotten? The early church agree that the Son is begotten by the Father. 


  "To beget is to produce an offspring of same nature."


The Father has no beginning that is why the Son has no beginning (like begets like).

Pre-Nicene Church Father Tertullian explains the begetting this way:


  God made [the pre-existent] Logos [a] second [person] to himself by agitating it within
  him.


St. Athanasius explains the begetting in this way:


  The Son is not "from nothing" but "from the Father".


How is the Son begotten?
Most theologians do not speculate as far as your question searches, but then again I think it is a valid question. The idea of the Son being begotten from the Father does not carry a connotation of time being that it was before time existed. Your question is therefore what connotation is expressed by 'begotten' if time is irrelevant? It would seem that the connotation of 'some essential filial' aspect of the Trinity is implied, so that in some sense the Father produces the Son, yet not independently of by his will alone. Without the Son there would be no Father, as God is Father, Son and Holy Ghost; eternally past, eternally present and eternally future, always and forever the same.  This may seem like a 'non-asswer' as it deals with the unknown but if we appreciate it as a basic manifestation of the triune relationships between the three persons of the unchanging Godhead, it naturally helps us understand that as God approaches time and his creation, he sent his Son and not himself. I mean the Father sent the Son to save us through an incarnation because the Father begets the Son and not viva-versa. The Spirit does not send the Son, the Son does not beget the Father, rather the Father begot the Son and both the Son and the Father sent the Holy Spirit. 

The relational aspects within the three persons of the Trinity our incomprehensible to us as they stand outside of time because we only live and understand a world in time. We can only really perceive its meaning within the time that God has entered to make himself known, the gospel. So how is the Son produced? Eternally, we can't know the inner workings of the Trinity beyond its basic definitions and assertion, but as the Father does eternally produce the Son outside of the concept of time, the Father has partly manifested what that means 'in time' by the incarnation.