What is this other begetting about?
"The Fathers make a distinction between eternity, which is proper to God, Father, Son and Holy Spirit, and to God alone,
What is their distinction between God's eternity and the Father, Son and Holy Spirit's eternity? "aeon"
This applies to some nebulous claims about emanations from God. The one sided eternity is just that. Of course this discussion involves the concept of time as a persistence of existence. That's the only workable concept of time that can be used here.
"Nor is the Father's person the Holy Spirit. The One God is tripersonal, three hypostases--the Father, the Son and the Holy Spirit--one ousia."
My understanding of what a person is and the use of the words spirit and Holy Spirit in scripture, lead me to believe that the spirit represents the person. The person of God is the Holy Spirit. Some think spirit means life giving force, but that doesn't stand up to reality.
I shall address your points in order from easiest to most difficult.
The aeon, as the Fathers use the term is not an emanation from God, but a feature of His creation: spiritual or noetic beings, the angels and Man are intended to be permanent. Unlike Him, they have a beginning, like Him, but unlike other created things, trees, rocks, even animals, they have no end.
I think you have failed to divide my sentence correctly, the distinction is between the eternity of God, Father, Son and Holy Spirit, One God, and the aeon of created spiritual (or more properly noetic) beings.
Your understanding of personhood is not congruent with that shared by the Father, who equated the Latin 'persona' with the Greek 'hypostasis', if you see only the personhood of the Spirit, and do not see the Father and the Son as likewise persons.
Finally, you ask what is the 'other' (I presume) eternal begetting of the Son from the Father. To paraphrase St. Gregory the Theologian, tell me first what is the unbegottenness of the Father, and what the procession of the Spirit from the Father, and I shall answer, and we will both be struck with madness for prying into the secrets of God.