Why does the Son have a source of existence?

You can see the answer to how this is possible in an example from creation. A yeast cell is about to have offspring by budding. (Here, I will only speak of one chromosome's DNA.) The original DNA is double-stranded. When it replicates, both original strands separate and make copies of the other strand on themselves. Now there are two double-stranded DNAs that have the original strand together with a new strand. Can you really say which one was first? One of these double-stranded DNAs will end up moving into a newly-formed membrane with cytoplasm from the original cell. So now you have a cell that was begotten, but since it still has half of the original DNA and cytoplasm from the original cell, can you really say that it wasn't there in the beginning? The fact that it's moving into a newly-formed membrane (like an incarnation) is new, but both cells have part of the original DNA in them, as well as an exact copy of the other strand. (Sort of like, "I am in the Father and the Father is in me.") Therefore, you can have a cell that is begotten, but it really was part of the original. In fact, you can say that one cell came from the other. This would be like Jesus saying he came from the Father, and it has a greater meaning than just being sent--it means, literally, coming forth from.

Origination (source) is a property of a person not of nature.

The Father, Son and Holy Spirit are equal in nature. They have the same nature but not the same person.

In regards to their person-hood, each of them are distinct rational individuals persons.

According to Christianity at large, there is only one God in three persons: while distinct from one another in their relations of origin as the Fourth Lateran Council declared:

"it is the Father who generates, the Son who is begotten, and the Holy Spirit who proceeds") and in their relations with one another, they are stated to be one in all else, co-equal, co-eternal and consubstantial, and "each is God, whole and entire"(source)

If God's nature includes being without origin how is it his Son, with an origin, is equal to that nature?

Nature doesn't have a personal property such as paternity, filial, and spiration. These hypostatic (personal) property belong to the three divine hypostases not their common essence (h*m* ousia). This is why Arius rejected h*m*ousia in Nicene council because he can't see how the Son who has an origin from the Father is equal with the Father. This Trinitarian issue still exist from Nicaea (325) until it's solved at Fourth Lateran (1215). To shed a little bit history let us visit the Fourth Lateran Council:

We condemn, therefore, and reprobate the book or tract which Abott Joachim published against Master Peter Lombard concerning the unity or essense of the Trinity, calling him heretical and insane because he said in his Sentences that the Father, Son, and Holy Ghost are some supreme entity in which there is no begetting, no begotten, and no proceeding. Whence he asserts that he (Peter Lombard) attributed to God not so much a trinity as a quaternity, namely, three Persons and that common essense as a fourth, ....

The Fourth Lateran Council (1215), Canon 2.

Here, Joachim accused Lombard for inventing a fourth person of Trinity namely the divine essence. Lombard was exonerated and Joachim was condemned. This is how Fourth Lateran explains the Orthodoxy of Lombard in the same canon:

But we, with the approval of the holy and general council, believe and confess with Peter (Lombard) that there is one supreme entity, incomprehensible and ineffable, which is truly Father, Son, and Holy Ghost, together (simul) three persons and each one of them singly. And thus in God there is only trinity, not quaternity, because each of the three persons is that entity, namely, substance, essense, or divine nature, which alone is the principle of the universe and besides which there is no other. And that entity is not the one begetting or the one begotten or the one proceeding, but it is the Father who begets, the Son who is begotten, and the Holy Ghost proceeds, in order that there may be distinctions in the Persons who unity in the nature. Though, therefore, the Father is one (being), and the Son is another, and the Holy Ghost is another, yet they are not different (non tamen aliud); but that which is the Father that is the Son and the Holy Ghost, absolutely the same, since according to the Orthodox and Catholic faith they are believed to be consubstantial. For the Father begetting the Son from eternity imparted to Him His own substance, as He Himself testifies: "That which my father hath given me, is greater than all" (John 10:29). And it cannot- be said that He gave to Him a part of His substance and retained a part for Himself, since the substance of the Father is indivisible, that is, absolutely simple. But neither can it be said that Father in begetting transferred His substance to the Son, as if gave it to the Son without retaining it for Himself, otherwise He would cease to be a substance. It is evident, therefore, that the Son in being begotten received without any diminution the substance of the Father and thus the Father and Son as well as the Holy Ghost proceeding from both are the same entity.

The Fourth Lateran Council (1215), Canon 2.

The Son who has a hypostatic origin begotten from the Father receives the same nature that His Father has. This way the Son is distinguishable from the Father hypostatically but not essentially. Just as a human father can be distinguished from a human son hypostatically but not essentially. The different is a human father can only share a portion of his nature to his son this is why a son is like (in Greek h*m*ios) his father. While the Father share the same nature (h*m* ousios) in its totality not partial to His Son. This is why in term of essence the Son is identical with the Father, h*m*ousia.

Arius can't see how the Begotten Son can be equal with the Unbegotten Father. Lombard developed a clarity between a hypostatic property of Unbegotten belongs only to the Father and His essence which doesn't have any hypostatic property. The divine essence clarified by Lombard is neither unbegotten nor begotten because those are hypostatic properties not essential. This is why the divine essence shared by the three divine persons is anhypostatic it has no personal property. In modern view of Reformed Trinitarianism developed by a Dutch American Cornelius Van Til he revived a view that God is three persons and one person referring to the divine essence as a hypostasis which the Fourth Lateran condemned. We see how Nicene faith, the faith of the Apostles is kept through present time through the Fourth Lateran Council faithfully that the divine essence is not a person.

In NME culture, the firstborn, especially the firstborn son, was very special indeed. He had privileges and perquisites subsequent children did not. Interestingly, one of the titles for Jesus is

"Firstborn of all creation" (Colossians 1:15).

This expression does not mean Jesus was created; it does mean that He bears the honorific title of Firstborn, with all the privileges that accrue from being so named. Moreover, He is the creator of all things (ibid., v.16); He existed before all things and holds all things together by the word of His power (ibid., v.17); He is the head of the church, His body; He is the beginning, the firstborn from the dead; He has the first place in everything (ibid., v.18); and all the fullness of Deity dwells in Him (ibid., v.19).

Perhaps the greatest verse to indicate the equality of Jesus with the Father (besides Jesus' own words, "He who has seen me has seen the Father. The Father and I are one person," John 10:30; 14:19; 17:21) is Colossians 2:9,

"For in [Jesus] all the fullness of Deity dwells in bodily form."

This verse does not say that Jesus became Divine when He came to earth as a baby, born of the virgin Mary. He was God the Son in eternity, and He never ceased being God the Son, even during the 33 years of His self-emptying life on earth as the God-Man, Jesus Christ (see Philippians 2:7 ff.). Moreover, He will never cease being God, albeit as God in the flesh-and-bone body of a man.

Within the triune Godhead, there is one shining star, and that is Jesus Christ, the only begotten Son of God. No one preceded Him; no one followed Him. He is sui generis, as the French say. While some Roman Catholics will disagree with me on this point, the eternal Son of God, the second person of the trinity, was neither conceived nor born; He simply is:

"Jesus Christ, the same yesterday, today, and forever (Hebrews 13:8)."

In eternity past, the Son was the delight of His Father, and the Father of the Son. In other words, they loved one another, and always will. Moreover, each person in the Trinity seems to have a unique, yet overlapping, role to play. Concerning the Father and the Son, Jesus said,

"'All things have been handed over [or delegated] to Me by My Father; and no one knows the Son except the Father; nor does anyone know the Father except the Son, and anyone to whom the Son wills to reveal Him'" (Mt 11:27; Lk 10:22).

While your question bypasses the role of the Holy Spirit, He too has been delegated a role; namely, to reveal Jesus to people by removing the blindness from their spiritual eyes and by imparting Jesus' words of life to them, first, when they are first born from above, and second, as they grow in Christ. Christ, in other words, is formed in us by the unseen work of the Holy Spirit of God during the lifelong process of sanctification.

The Trinity is indeed a mind-blowing doctrine, neither easily explained nor easily understood. It is, however, part of the bedrock of the Christian faith. The only acceptable sacrifice for sin had to be both fully God and fully man, the infinitely perfect, spotless, and holy Lamb of God. Anything less would not suffice to put away sin by the sacrifice of Himself (Heb 9:26).

You are confusing the Deity Jesus with the man Jesus.

When the Angel Gabriel announced to the Virgin Mary that she would become Pregnant and bear a son, he told her that she would be overcome by the Holy Spirit.

Luk 1:35 KJV

And the angel answered and said unto her, The Holy Ghost shall come upon thee, and the power of the Highest shall overshadow thee: therefore also that holy thing which shall be born of thee shall be called the Son of God.

The Holy Ghost himself placed the seed of God into Mary's womb which combined with Mary's seed then grew into the baby Jesus.

Because Jesus had both the seed of man which he inherited through Mary his mother and the seed of God which the Holy Spirit placed in the womb of his mother Mary, Jesus was both man and God.

This is no different than the fact that you are the son of both your mother and your father, and have characteristics of both of them.

Mat 1:18 KJV

Now the birth of Jesus Christ was on this wise: When as his mother Mary was espoused to Joseph, before they came together, she was found with child of the Holy Ghost.

Mat 1:20 KJV

But while he thought on these things, behold, the angel of the Lord appeared unto him in a dream, saying, Joseph, thou son of David, fear not to take unto thee Mary thy wife: for that which is conceived in her is of the Holy Ghost.

That seed from the Holy Ghost is eternal therefore the Deity part of Jesus is also eternal.

John 1:1,14 is the classic proof text of the eternality of the Son.

in the beginning was the Word, and the Word was with God, and the Word was God. Through Him all things were Made, and by him was nothing made that was made.

John 1:14 goes on to say that this is Jesus.

Furthermore the creeds are very clear that he was begotten not made, which precisely is there to address the whole subordinationism debate. Basically, the creeds work out in more precise detail how it is that the Son is not subordinate to the Father, by nature of his eternal begottenness.

To go too much further would really be to stumble into book territory, which is beyond the scope of the site.

... his Son, with an origin,....

This is one of the classic misconception of Trinity. Jesus is not begotten in a particular point in time by The Father. As you have mentioned Jesus is eternally begotten. That is, in the very nature of God's existence, Jesus was/is/will-be begotten by the Father for eternity. This is same as how holy spirit was/is/will-be proceeding from the father (and the son). That is being eternally begotten is not an event in the past, rather the state of existence (being) of second person of the Trinity in the Trinity.