Why are we called "the Jews"? Did Hashem pick out this name for us? What does the word "Jew" mean?
It's from "yehudah", because most of the Jews nowadays are from the shevet of yehudah.
The name Jew actually comes from Judea, which is a place. The place of course is named after Yehuda. But importantly, in the age of Hellenization, the concept "ioudaios" (Jew) shifted from being an ethnic-geographic marker to a religious-cultural one. Thus, Judaism itself was Hellenized to become a religion where people or groups could join. (see Shaye Cohen, The Beginnings of Jewishness)
Jew come from the Name Yehuda in most languages it is something similar to Yehuda, it represents anyone who denies Avodah Zara.
We are called Jews from the time we returned from exile in Babylon. Since then, the Tribal system is over, and just one People exist: The People of Israel (the new Israel) the Jewish People. This People, although made up by the main stem of Judah, it constitutes of a few tousands from the other Tribes, and two thirds of the Levites, plus the properly converts from the Gentiles.