Part of the difficulty with this question is the fact that the word Jew does not occur in the Bible until 2 Kings 16:6 (KJV) and 2 Kings 25:25 in most other Bible translations. In those instances, the Hebrew word would likely be better translated “men of Judah.” The word Jew is much more commonplace in the books of Ezra, Nehemiah, and Esther. The usage of the word Jew in those books helps explain the origin of the word and why it was used.
The answer to the question “Who was the first Jew?” depends on what exactly is meant by the word Jew. Originally, God’s chosen people were known as the Hebrews. Later, after they settled in the Promised Land and formed a nation, they were known as the Israelites. The term “Jew” did not come into use until after the ten northern tribes were exiled to Assyria and Judah was exiled to Babylon. In the later stages of the captivity (Esther) and in the early stages of the return to the land of Israel (Ezra and Nehemiah), the tribe of Judah was dominant. The word Jew developed as a shortening of the word Judah. But the word Jew was used as a descriptor for more than just the tribe of Judah. The dominance of the tribe of Judah in the return to the Promised Land resulted in all of the Israelites, people from all 12 of the tribes, being referred to as “Jews.”
So, who was the first Jew? If by “Jew” we mean “Hebrew,” Abraham was the first Jew. If by “Jew” we mean “of the tribe of Judah,” Judah was the first Jew. If by “Jew” we mean “the first person in the Bible to be referred to as a Jew,” the nameless Jews in 2 Kings chapters 16—25 were the first Jews. Generally speaking, people today use the term Jew to refer to “a person who is of the chosen people of Israel.” With that in mind, Abraham should be considered the first Jew.