The terms Hebrew, Israelite, and Jew are used interchangeably in the New Testament, but not in the Old Testament.
Abraham was called a Hebrew (Genesis 14:13), which means “to cross over.” (He had crossed over the River Euphrates.) The blessings God promised His descendents were passed along to the children of Jacob whose name was later changed to Israel (Genesis 32:28). It was his name that was given to the “children of Israel” or the Israelites. One of Jacob’s sons, Judah, is the man from whom we get the term Jew. The term was not used until many years later (2 Kings 16:6 KJV).
Following King Solomon’s death, his realm was divided into northern and southern kingdoms, Israel and Judah respectively. Some members of the 10 tribes who lived within the southern kingdom chose to remain with Rehoboam, King of Judah (1 Kings 12:23-24). Others returned when Asa was Judah’s king (2 Chronicles 11:14-17), following the Assyrian deportation of the upper classes of the northern kingdom in 721 bc, and during the religious revival under Hezekiah. Another return occurred under Josiah (2 Chronicles 34:9), and representatives of all 12 tribes were present in the returns under Zerubbabel and Ezra, signified by the fact that Ezra “offered at the dedication . . . twelve he goats, according to the number of the tribes of Israel” (Ezra 6:17 KJV).
The term Jew undoubtedly began as a designation of one of the members of Judah as distinguished from the 10 tribes. But as the term Israel gradually began to refer to the kings of Judah (see 2 Chronicles 21:1-2 in connection with Jehoshaphat; 2 Chronicles 28:19 in connection with Ahaz), the term Jew began to include all who remained loyal to the Mosaic Law and the city of Jerusalem, whether they were descendents of one of the 10 northern tribes or the two southern tribes of Judah and Benjamin. The terms Israelites and Jews became synonyms. Note that the Jewish prophetess Anna, who lived in the temple, was from the tribe of Asher (Luke 2:36).
Along with the amalgamation of the 10 tribes with Judah and Benjamin, multitudes of proselytes were converted to Judaism as well. So even though the people we know as Jews today are not necessarily physical descendents of the tribe of Judah or even of Abraham (See the ATQ, Are today’s Jews the physical descendents of Abraham, Isaac, and Jacob?), they symbolically represent not just Judah but all 12 tribes. (It’s important to note as well that the Judaism that grew out of Judaism’s corporate rejection of Jesus Christ is not the same as that which existed at the time of His coming. See the article, Is Judaism today basically the same as Judaism at the time of Jesus Christ?)