Jew, Juif, Jude, Yid etc are all Europeanised versions of Yehudi, which refers to someone from the tribe or region of Yehudah, Judah. How did the name of one of Jacobs sons and the Twelve Tribes come to denote the whole people of Israel? It happened late in the biblical period. After the death of King Solomon, his kingdom split into two parts; the Kingdom of Judah which comprised the tribes of Judah and Benjamin, and the Kingdom of Israel, made up of the other Ten.
Around 700 BCE Sennacherib, King of Assyria, carried away the Ten Tribes into exile and they were effectively dissolved (though many cranks throughout history have claimed to find, or to be, their descendants). From then on, Yehudi came to mean the collectivity of Jews in the Land of Israel.
This usage received a new twist when Nebuchadnezzar, King of Babylon, exiled the remaining two tribes in 586 BCE. The exiled people were known as Yehudim. For example, in the Book of Esther (2:5), Mordecai is called an ish Yehudi, and then were told that he was a scion of Benjamin. Rashi explains that even so, the Megillah calls him a Yehudi because Judah was the larger of the two tribes.
The name stuck.
Many have linked the reasons for Judahs naming with some essential feature of the Jewish people. This time I will thank (odeh ) God, his mother Leah says (Genesis 29:35). To be a Jew is to know how to give thanks.