The idea of a human or group of humans being God's son is not uncommon in the Hebrew Tanakh ("Old Testament"). For example, in Exo. 4:22 (cp. Hos. 11:1), Yahveh commands Moses to say to Pharoah,
Thus said Yahveh, "Israel is My son, even My firstborn."
The motif of the nation of Israel being God's child is reiterated in various other books of the Tanakh.
Speaking of Yahveh, Moses asks the Israelites (Deu. 32:6),
"Do you thus requite Yahveh, O' foolish and unwise people? Is He not your father who bought you? Has he not made you and established you?"
Similarly, in Mal. 2:10, it is written,
Have we not all one father? Has not one God created us? Why do we deal treacherously every man against his brother, by profaning the covenant of our fathers?
But, of course, none of the Israelites were ever actually begotten by God the Father. They weren't literally God's sons or daughters. Rather, they were sons and daughters by adoption (Greek ἡ υἱοθεσία; cp. Rom. 9:4). They were called sons and daughters, but all had actual human parents of whom they were literally begotten as sons and daughters.
Accordingly, some might insist that Jesus, despite being called the "son of God," is no different than any other Israelite being called God's son. However, we should note that Jesus distinguishes his relationship with the Father to that of everyone else's (i.e., other Israelites) relationship to the Father.
For example, in John 20:17, it is written,
Jesus says to her, "Do not touch me, for I have not yet ascended to my Father, but go to my brothers and say to them, 'I ascend to my Father and your Father, to my God and your God.'"
Why didn't Jesus simply say, "I ascend to our Father and to our God"? Instead, he explicitly distinguishes his relationship with the Father to theirs. There's also no doubt that Jesus' words were understood as something more than the historical status quo. Jesus' statement that God was his Father was understood as blasphemous, worthy of death.
For example, in John 5:17-18, it is written,
17 But Jesus answered them, "My Father worketh until now, and I work." 18 Therefore the Jews sought to kill him even more, because he not only had broken the Sabbath, but he also said that God was his Father, making himself equal with God.
As Augustine wrote,1
He saith not, Our Father: in one sense, therefore, is He mine, in another sense, yours; by nature mine, by grace yours. “And my God, and your God.” Nor did he say here, Our God: here, therefore, also is He in one sense mine, in another sense yours: my God; under whom I also am as man; your God, between whom and you I am mediator.
Non ait: Patrem nostrum: aliter ergo meum, aliter vestrum; natura meum, gratia vestrum. Et Deum meum, et Deum vestrum. Neque hic dixit: Deum nostrum: ergo et hic aliter meum, aliter vestrum; Deum meum sub quo et ego homo sum, Deum vestrum inter quos et ipsum mediator sum.
So, Jesus' identity as the "son of God" is indeed different from ours as Christians as well as the Israelites before. Jesus is the son of God by nature, both physically and spiritually begotten by God the Father. Whereas, Christians are spiritually begotten by Holy Spirit via regeneration (being born again), but obviously, we are physically begotten by our human parents.
John of Damascus wrote,2
Since, indeed, he participated just as we ourselves do in blood and flesh and became man, while we too through him became sons of God, being adopted through the baptism, he who is is by nature (φύσει) Son of God became first-born among us who were made by adoption and grace sons of God, and stand to him in the relation of brothers. Wherefore he said, "I ascend to my Father and your Father." He did not say "our Father," but "my Father," clearly in the sense of Father by nature (φύσει), and "your Father," in the sense of Father by grace (χάριτι).
Γεγόναμεν δὲ καὶ ἡμεῖς δι' αὐτοῦ υἱοὶ θεοῦ υἱοθετηθέντες διὰ τοῦ βαπτίσματος· αὐτὸς ὁ φύσει υἱὸς τοῦ θεοῦ πρωτότοκος ἐν ἡμῖν τοῖς θέσει καὶ χάριτι υἱοῖς θεοῦ γενομένοις καὶ ἀδελφοῖς αὐτοῦ χρηματίσασι γέγονεν. Ὅθεν ἔλεγεν· «Ἀναβαίνω πρὸς τὸν πατέρα μου καὶ πατέρα ὑμῶν.» Οὐκ εἶπε· πατέρα ἡμῶν, ἀλλὰ «πατέρα μου», φύσει δῆλον, καὶ «πατέρα ὑμῶν» χάριτι.
In summary, the reason why Jesus' admission of being the "son of God" was considered blasphemy by the chief priest was because the chief priest acknowledged it (just as others did in the Gospel of John) as a claim that Jesus was likewise God in nature, and of course, the chief priest didn't believe that claim was true. I'm not exactly sure it was Jesus' latter claim about coming upon the clouds of heaven that was the impetus for the chief priest's accusation of Jesus committing blasphemy. In my opinion, it's likely the former admission of Jesus admitting (or rather, not denying) he is the son of God. It's unfortunate that even many Christians neglect to recognize the import of individuals claiming Jesus (and he himself admitting) to be the son of God.
1 Commentary on John 20:17: English | Latin
2 Exact Exposition of the Orthodox Faith, Book VI, Ch. 8: English