In the Bible, the people of Israel are known as the people of Israel, Sons of Israel, Israelite.
What is the source of the name Jewish? When did it start?
The people of Israel are named Jewish after Judea, the fourth son of Jacob.
The twelve sons of Jacob started their own families, multiplied and became large families - Tribes. Before Jacob passed away, he blessed each and every son of his, each and every head of a tribe.
Jacob's blessing to Judea was that his brothers will acknowledge his greatness, his leadership and will answer to him as the leader of the Sons of Israel. This is why the people of Israel are called 'Jewish'.
How did Judea earn that honor?
Judea had earned this honor thanks to his modesty and humility. At a certain event, he could have had someone else take blame while he remains 'clean'. Judea acted otherwise.
He admitted that he was the responsible - even though that could have destroyed his image and reputation.
Thanks to that specific event of leadership he had earned the leadership of the Jewish people. King David came from the lineage of Judea and so will the Messiah.