In bible times the phrase "the Son of God" was understood as a claim to the nature of God, not a claim to be a lesser being:
"The Jews answered him, We have a law, and by our law he ought to die, because he made himself the Son of God." John 19:7
Ron Rhodes commments:
"Ancient Semitics and Orientals used the phrase "son of . . . " to indicate likeness or sameness of nature and equality of being. Hense, when Jesus claimed to be the Son of God, His Jewish contemporaries fully understood that He was making a claim to be God in an unqualified sense. Benjamin Warfield affirms that, from the earliest days of Christianity, the phrase "Son of God" was understood to be fully equivalent to God. This is why when Jesus made His claim, the Jews insisted, "We have a law, and according to that law he must die, because he claimed to be the Son of God" (John 19:7). Recognizing that Jesus was identifying Himself as God, the Jews wanted to put him to death for committing blasphemy (see Leviticus 24:16)." Reasoning from the Scriptures with the Jehovah's Witnesses. p. 243
If the above is correct, and the phrase "the Son of God" also meant that Jesus was by nature God, then we would expect that He would be called both "God" and "Son of God" in the scriptures:
"In the beginning was the Word, and the Word was with God, and the Word was God." John 1:1
"And Thomas answered and said unto him, My Lord and my God." John 20:28
"But these are written, that ye might believe that Jesus is the Christ, the Son of God; and that believing ye might have life through his name." John 20:31
(Keep in mind that when Trinitarians claim the Jesus is God, they are not saying that Jesus is the person of the Father, but that Jesus is God by nature along with the Father).