Why are we called "the Jews"? Did Hashem pick out this name for us? What does the word "Jew" mean?
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)
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.
The earliest reference to the term Jew (יהודי in Hebrew is found in Esther 2:5 where it has Mordechai
האִ֣ישׁ יְהוּדִ֔י הָיָ֖ה בְּשׁוּשַׁ֣ן הַבִּירָ֑ה וּשְׁמ֣וֹ מָרְדֳּכַ֗י בֶּ֣ן יָאִ֧יר בֶּן־שִׁמְעִ֛י בֶּן־קִ֖ישׁ אִ֥ישׁ יְמִינִֽי:
5 There was a Judean man in Shushan the capital, whose name was Mordecai the son of Jair the son of Shimei the son of Kish, a Benjamite,
a Judean man: because he was exiled with the exile of Judah; all those who were exiled with the kings of Judah were called יְהוּדִים [Judeans, Jews] among the nations, even if they were from another tribe.
Thus we see that all members of our religion, from that time on were called by the term that is currently translated as Jews because of the exile from the Kingdom of Judah.
In the Hebrew Bible, the Jews are called B’nei Yisrael, Israelites. After the reign of Solomon, the unified twelve tribes split in two, the northern kingdom was called Israel, and the southern kingdom was called Judea, since this southern kingdom was mostly comprised of people from the tribe of Judah. These people were called Judeans. The name Judeans is shorted for Jews. In 722 BCE, the people of Israel were defeated by the Assyrians and were sent into exile, becoming the famous “Lost Ten Tribes,” Judea itself became exiled by the Babylonians in 586 BCE. But in 536 BCE, the exiles reestablished their country rebuilt its temple and thrived until the Roman destruction. And again, after 2000 years of exile, they reestablished the modern State of Israel.
In short, Jews are called Jews because it is short for the name Judeans.