In other words, how weird was it at the time for Jesus to proclaim himself not only a prophet, but the son of God? Were miracle-workers a commonplace thing at all? Does Jesus just happen to be the one self-described prophet who got popular, or was he a truly unique case?
Obviously Jesus' claims, and the records of his life in the Gospels, are remarkable. He is historically treated to be distinct from other cases of prophets because of his self-described status as the son of God- it's one thing to be more connected to the divine, and it's quite another to literally be the divine on Earth. Obviously this claim resulted in him being crucified. But was there a cultural precedent for "prophets" at the time?
edit: a further clarifying question would be: did other people proclaim themselves to be the Messiah at the time? Is Jesus just the one guy doing this who saw the most popularity?
Great question!
First thing to note is that 'son of God' can mean many things and doesn't necessarily mean the divine on earth. Israel as a whole is referred to as the son of God in Hebrew scripture, and it's used to describe the pious, kings and angels. Jesus himself repeatedly refers to his followers as being/becoming sons of God. In Rabbinic literature nearer Jesus' times it's used of charismatic miracle-workers in a way that sounds like it could be Jesus e.g.
The whole world is sustained on account of Hanina, my son, but Hanina my son is sustained by one kab of carob from one Sabbath eve to another
In a religion as emphatically monotheistic as Judaism, I don't think we can read references to 'son of God' as meaning the person is God.
It's not at all clear that Jesus was thought of at the time as 'that guy who said he's son of God' and very little reason to think he was thought of as claiming to be God (after all he said followers would also be sons of God so unclear he's claiming a unique status). At most he's accused of acting as if he has divine authority but that can be seen in terms of a 'special connection'. He's more set apart in John's Gospel but that's later with major Platonist influences and questionable as a report of what happened. The Gospels say he was crucified for claiming to be 'King of the Jews', not for claiming divinity.
The most clearly attested elements of Jesus' life are as a miracle-worker and faith-healer preaching an imminent apocalypse: that definitely fits within the milieu of the time. There were also various claimed Messiahs around the time (some of whom led armed revolts and were killed). Just to be explicit, claims to be the Messiah are definitely not claims of divinity in Judaism (though the Messiah is referred to as 'son of God'.
EDIT: main source for specifics above is
Geza Vermes From Nazareth to Nicea
but I'd also recommend
Charles Freeman A New History of Early Christianity
Bart D Ehrman The Triumph of Christianity
Dositheus the Samaritan made messianic claims around the same time as Jesus.
His alleged protege Simon the Magus did as well. Simon made faith healing claims and said he was the son of god, and founded his own churches, with his own cosmology. Some Jews thought he was the Messiah. Naturally he clashed with several apostles. He was called "the father of heresies," but also may have been correctly termed the father of gnostics. Christians have written extensively to discredit Simon the Magus, starting with references to him in Acts that were meant to distinguish him from Jesus. And this debate goes on today.
You can read about this controversy here.