The answer to this question can be determined by seeing whether the Bible speaks of the Son being the Son of God eternally, or only becoming the Son of God at a particular point in time - in the case of this question, at the time of Christ's resurrection from the dead. As there is also a related question as to whether he could have become Son of God at his incarnation, I start with a quotation from a scholar on this point:
"There is the clear evidence for an eternal sonship afforded
throughout John's Gospel. ...It is implied, for example, in John 3:16,
where the greatness of God's love lies precisely in the fact that he
gave his Son to incarnation, humiliation, pain and death. He was Son
at the point of giving; that is, at the point of incarnation
(sending). Calvary is the climax of the giving, not its commencement.
Equally, the eternity of the sonship lies behind John 17:1, 5:
'Glorify your Son... with the glory I had with you before the world
began.' This clearly implies that his pre-existent glory was a glory
he had as Son. John 6:46 speaks to similar effect: 'No-one has seen
the Father except the one who is from God; only he has seen the
Father.' It is difficult to believe that this refers to some vision
enjoyed by Jesus after the incarnation. It is much more natural to
take it as referring to something pre-temporal, especially in view of
Jesus' description of himself a few verses later as 'the bread that
comes down from heaven' (Jn.6:50). He comes to earth both as the Son
and as the bread of life.
The following conclusions, then, seem safe. First, monogenes says
nothing about origins because the Son is unoriginated. Secondly, it
emphasizes the uniqueness of Jesus' sonship. Thirdly, this uniqueness
consists in four things: he is an object of special love, he is the
Father's equal, he is the Father's likeness and he is an eternal, not
an adopted. Son." (The Person of Christ p74, Donald Macleod,
Inter-Varsity Press, 1998)
Another vital point regarding this question was raised in one set of comments where it was claimed that the Son is subordinate to the Father, citing Christ delivering the kingdom to God. "The reason for dissolution of the powers and the world is because of end of the world; He will not reign after he has destroyed his enemies, bec theres no need for it. He will not have the need to be Lord, and Son which are merely temporal offices... Sonship exists only as long as the purpose of judging the workers in the vineyard, as the heir of the Father God."
This matter of subordination needs to be faced up to, and over four pages, citing many scriptures that both show the Son's apparently delegated authority, and others showing the equality of the Son with the Father, the author above makes these points:
"The presence of subordinationist material should in fact be seen as a
tribute to John's historical credibility. It embarrasses his main
thesis (the divine grandeur of Christ), yet he resists the temptation
to suppress it... we must pay attention to the purpose of these
subordinationist passages. Jesus was vulnerable to the charge that he
was concerned only with his own glory. He appeared to make himself
equal with God; he performed miracles without any specific invocation
of God; he legislated by saying simply, 'I say to you!' Hence the need
to remind his contemporaries that he was in the world to reveal and
glorify the Father and to do his will. But just there lay the paradox:
to reveal the Father he had to reveal himself. ('Anyone who has seen
me has seen the Father', Jn.14:9). Hence the two kinds of material:
the material that reveals the Son, and the material that obscures him.
There is no way of resolving the difficulty, if only because the
Father is as determined to glorify the Son as the Son is to glorify
the Father. This is why Jesus himself can say, 'Now is the Son of Man
glorified and God is glorified in him' (Jn.13:31). Hence, too, the
portrayal of even the cross itself as a 'lifting up' (hypsosis:
Jn.3:14; 12:32; 12:34). (Ibid. pp 75-78)
It is mentioned that the question of subordination resulted in language used by the early church fathers that allowed a vocabulary of subordinationism to flourish, but that N.T. subordinationism is federal, not ontological. All of that is too complex to do justice to here; a distinct question needs to be raised to deal with the matter. I include a couple of points about it here because it is directly related to the question of whether the Son of God became the Son either at the time of his resurrection, or at his coming into the world as the Son of Man.
Now, the main point: What the scriptures state about the Father and the Son shows the intimacy of relationship of their unique relationship in one, Holy Spirit. All that would declare the Son to have only entered into a relationship of Son with God the Father at some point in time, robs the Deity of the intimacy of relationship between Father and Son, in One Holy Spirit. This triune relationship
"is not a solitude. This relationship is (as the three Luke passages
show [7:12; 8:42; 9:38]) not a question of nature. It is not of
nature, of gender, of natural conception, of flesh and blood, of
carnal connection. It is a relationship of person... the everlasting
relationship between Father and Son, in One Spirit." (The Only
Begotten Son of God, pp 14-15, Nigel Johnstone)
The author (above) goes on to show the eternal begetting of the Son, saying of those who wish to attack that biblical truth:
"...they will have (only in their miserable darkened mind) three gods
with no divine relationship conjoining them. Or one god who is
incapable of any action whatsoever - for he is, utterly and incurably
: alone. ... John 5:25 records Jesus' words, in context: 'Verily,
verily, I say unto you, The hour is coming, and now is, when the dead
shall hear the voice of the Son of God : and they that hear shall
live. By the voice of the Son of God, shall they live. For as the
Father hath life in himself, ; so hath he given to the Son [of God] to
have life in himself.'
Note the article in v25 and again in v26. But note the absence of
article in the following verse : 'And hath given him authority to
execute judgment also, because he is son of man', John 5:27.
The Authorised Version translators supply a definite article in the
English translation and then fail to put it in italics, contrary to
their own rules, for there is no article in the Greek original of the
Received Text, something noteworthy in a language which so frequently
presents it.
Jesus identifies 'the' Son of God, then identifies 'the' Son but
presently only 'son of man' thereafter. 'He' - that is 'the' Son - is
'son of man'. I say again, verse 26 is categorically not about
incarnation, but 27 is clearly so.
The power of his voice (surely no man is so mad as to assert that this
refers to the sound that comes from Jesus' human throat!) is due to
his Sonship in Deity. And Jesus explains the reason for such power :
it is due to the life that is of the Father being in the Son,
independently - which is a begetting - the reason that the dead hear
his voice." (Ibid. pp 30-32)
Having supported the biblical claim that the Son of God has eternally been Son, the Romans 1:2-3 text is seen to be correctly translated (e.g. YLT and other translations) as "marked out Son of God in power, according to the Spirit of sanctification, by the rising again from the dead,) Jesus Christ our Lord". He was never made Son by the resurrection, or even by the incarnation. It speaks of the everlasting relationship between Father and Son, in One Spirit.