What did the word “Son of God” mean to the Jewish people in Jesus’ time?
is there any historical evidence about what that meant?

I am afraid you that I am not an expert on this question by a
long shot. I will provide at least a hint of an answer, but suggest you
continue your research on this question elsewhere.

To the Jew of the first century, the phrase “Son of God” would
probably have had as much of a Greek feel to it than a Jewish feel to it.
In Greek mythology, the idea of being a son of the gods was taken almost
literally, as they had myths of gods copulating with humans, producing
demi-gods and so forth. For the Jewish mind, with such a strong emphasis
on monotheism, the use of the phrase Son of God applied to a human may
well have been labeled blasphemy if it had implications of deity attached
to it.

Having said that, the title son of God (little s; not implying
deity) is found in the Old Testament, at least indirectly. In the context
of the Old Testament, the Hebrew word translated “son” in the English has
an implication of connection to or intimate relation with God, as opposed
to implying deity or equality with God. Examples of the use of a phrase
similar to son of God (without implication of deity or implying being
Messiah) are found in Job 1:6, Job 2:1 (referring to angels), 2:1,
Deuteronomy 14:1 (“you are children of the Lord your God”) and other
passages.

Perhaps the most interesting reference to a “son of God” is in
Psalms 2:7. Here, God is being quoted by David as saying; He said to
me, “You are my Son today I have become your Father.” Although the direct
phrase son of God is not being used here, the passage certainly is
implying someone is the son of God. This is the only Old Testament
passage that I know of (and I have already said I am not an expert) in
which a phrase similar to son of God is used to apply to what is clearly a
messianic figure. With this one example, I would say that for the Jew,
the title Son of God as a reference to the Messiah would not have been a
familiar thing at all.

What was very familiar to the Jews was the phrase “Son of
Man.” This description is used by God to call Ezekiel in the Old
Testament. For example, Ezekiel is called “Son of Man” in Ezekiel 2:1 and
Ezekiel 8:5 and in several other passages. In this context, the
designation son of man is not implying deity, but rather God uses it to
point out the prophetic office of Ezekiel and his special role as
spokesperson to Israel in exile. Perhaps even more interestingly, the
phrase is used in Daniel chapter seven to describe what may very well be a
heavenly, prophetic vision of the Messiah:

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I saw in the night visions, and, behold, one like the Son of man came with
the clouds of heaven, and came to the Ancient of days, and they brought
him near before him.

So, I would say that, on balance, the use of the phrase “Son
of God” to describe the Messiah–God in the flesh was a new thing to the
Jews in Jesus’ time. It is worth noting in this connection that Jesus was
the first to use the word “Father” to refer to Jehovah, the God if
Israel. Jesus called God Father in almost every case that he referred to
God. It is not surprising then, that when God spoke of Jesus on the mount
of transfiguration, he said, “This is my son, of whom I am well pleased.”

In summary, in the Old Testament, the idea of being a son of
God was not unknown, but except with rare exceptions (Psalms 2:7 and
Daniel 7:13, in both of which the actual phrase “Son of God” is not used)
it had implications of being one in an intimate relationship with God, not
deity. Therefore, when Jesus came referring to the God of Israel as his
Father, and himself as the Son of God, that was disturbing and
revolutionary for most of the Jews who heard Jesus.

John Oakes