Let me put this in context by examining four closely related phrases.
Son of Man
"Son of man occurs in three different ways:
- A title for humanity generally Num 23:19, 1 Sam 5:12, Job 25:6, Ps 8:4, 56:2
- Title or form of address for just two of the prophets Eze 2:1ff (about 80 or 90 times) Dan 8:17 (only once)
- A title for Messiah in Dan 7:13 and often used in the NT
In most instances this appears to emphasis mankind's mortality.
Sons of God
"Sons of God" is used in some highly debated ways
- Humans who are servants/disciples of God Gen 6:2-4, Matt 5:9, Luke 20:36, John 1:12 Rom8:14, 19, Gal 3:26, Phil 2:15, 1 John 3:1, 2, etc.
- Non-earthly beings Job 1:6, 2:1, 38:7
Son of the Devil
The phrase "son of" is used as a title of character in places like John 8:44 (sons of the Devil), 1 Kings 21:13 (literally sons of Belial), Eph 2:2 (sons of disobedience).
Son of God
Apart from the above references to "sons of God" the phrase "son of God" appears to be almost exclusively reserved for Messiah (Mark 1:1, 5:7, 2 Cor 1:19, Heb 6:6, etc) but not exclusively. It is also applied to Adam, Luke 3:38. In the debatable reference in Dan 3:25, Nebuchadnezzar appears to be observing a person ("the son of God") with a particularly noble bearing.
Thus, as best I can determine, the title "son of" is more a statement of character that true origin. The exception might be John 5:18. Thus, Jesus' two titles are significant, "Son of God" and "Son of Man" about which John makes much (1 John 4:3, 2 John 7). Thus, "Son of God" appears to be simply a statement that the person is God-like in character.
The comment in Matt 4:3 is, I believe, deeply significant. Jesus was God (Phil 2:5) and so, could have easily made the stones into nourishing bread. Both Jesus and the devil knew this. Jesus wanted to be a "son of God" that depended on the Father for all things, especially as an example to us being "sons of God".