Where did the term "Jew" come from?

Two appendices in The Jewish Annotated New Testament touch on the issue:

Greek-speaking Jews in antiquity regularly referred to themselves as Ioudaioi. As an ethnogeographical term, best translated "Judeans," it designates the members of the ethnic group inhabiting the district of Judea, or their descendants wherever they may be. It translates the Hebrew term yehudim, which appears in the Hebrew Bible in books of the exilic period (2 Kings, Jeremiah, Ezra-Nehemiah, Esther). In the course of the last centuries BCE and the first century CE the ethnogeographical meaning of Ioudaioi receded, and a new religious meaning came to take its place. As a religious term, best translated "Jews," it designates people of whatever ethnic or geographical origins who worship the God whose temple is (or, after 70 CE, had been) in Jerusalem. Modern translations, including the New Testament translation used in this volume, usually take Ioudaioi as a religious term ("Jews") rather than as an ethnogeographical term ("Judeans"), although this translation is not always certain or correct.¹

And:

When Alexander the Great conquered the Persian Empire in the late fourth century BCE, one of the many lands he obtained was the small province, centered upon Jerusalem, called Yehud in Aramaic (Ezra 5.1; 5.8; 7.14). This name stemmed from the kingdom of Judah (Heb Yehudah)—the Southern Kingdom under the descendants of King David, who had ruled that land—after the separation from the Northern Kingdom of Israel in the late tenth century BCE, for more than three centuries before its occupation, first by the Babylonians (597–586 BCE), then by the Persians (539 BCE). Greek-speakers such as Clearchus of Soli (ca. 300 BCE) soon after began referring to this region as Ioudaia, "Judea," and its inhabitants as Ioudaioi, "Judeans."

. . .

As with any term of identity, dispute and/or confusion at times emerged in antiquity over who properly should be called Ioudaios. The problems began in the second century BCE, when an important development complicated the definition of the term Ioudaios. When the Hasmonean kings expanded Judean hegemony by conquering regions to the north and south of Judea—e.g., Samaria, Galilee, and Idumea—they imposed their laws on the native populations. As a result, many who previously had no ethnic or geographic connection to Judea became Ioudaioi, inasmuch as they resided on lands controlled by Judea and obeyed its laws. Yet, opinions varied regarding the extent to which one actually became a Ioudaios through such incorporation.

. . .

In recent years, some scholars have argued that translating Ioudaios with two terms—"Jew" for religious connotation and "Judean" for the ethno-geographic one—is anachronistic. . . . The term Ioudaios designated a person who was from or whose ancestors were from Judea, and for that reason worshiped the God of Judea.²

Summary

Because ethnicity and geography and religion were so closely associated in the ancient world, the term 'Judean' referred to the ethnicity, the geographic homeland, and the religion, all rolled into one, of the people group that came from the Kingdom of Judah.

Following the exilic period, the term eventually became semi-synonymous with 'Israelite' in its usage (since it could refer to people who were from an Israelite tribe other than Judah).

As the Judean ethnic group dispersed into other regions, and as the Judean religion became accepted by different ethnicities, the term slowly began to take on a broader meaning that could apply to any of those three aspects (ethnicity, homeland, or religion), without necessarily implying the other two. It's common for the word to be rendered as 'Jew' in English, most often in reference to the religion.

¹ Shaye J.D. Cohen, 'Judaism and Jewishness', p. 513.

² Joshua D. Garroway, 'Ioudaios', p. 524-525.