The term "Jew" is derived from the name of Jacob's fourth son, Judah--Yehudah, in the Hebrew--and may have originally applied only to Judah's descendents, who comprised one of the twelve tribes of Israel. On his deathbed, Jacob assigned Judah the role of leader and king--a prophesy that was fulfilled in 869 BCE when all twelve tribes submitted to the reign of King David of the tribe of Judah.
After the death of David's son, King Solomon, a civil dispute split the twelve tribes of Israel into two kingdoms: the Kingdom of Judah in the south, which included the tribes of Judah and Benjamin (and some Levites and priests) and was centered around the capital Jerusalem and the Holy Temple; and the northern Kingdom of Israel, which included the other ten tribes.
In the 5th century BCE, the Kingdom of Israel was conquered by Assyrian King Sennaherib, and the ten tribes were exiled and lost. The only remaining Israelites were the residents of the Kingdom of Judah, and the term "Yehudi" or "Jew" came to refer to all the Israelites, regardless of their tribal ancestry.
But there is also a deeper meaning to the name "Jew. The first individual to be called a Jew (Yehudi) in the Scriptures was Mordecai, of Purim fame. "There was a man, a Yehudi, in Shushan the capital, whose name was Mordecai . . . a Yemini" (Esther 2:5). The Talmud (Tractate Megillah 12b) asks on this: "He is called a Yehudi, implying that he descended from Judah; he then is called Yemini, implying that he is a Benjaminite!" Rabbi Jochanan responds: "He was a Benjaminite. Yet he was called a Yehudi because he rejected idolatry--and anyone who rejects idolatry is called a Yehudi."
The commentaries explain that the name Yehudah shares the same root as the Hebrew word hoda'ah, which means acknowledgement or submission. One who acknowledges G‑d's existence and submits to His authority--to the extent that he is willing to sacrifice his life for the sanctification of His name--he is called a Yehudi.
Hence Abraham is commonly referred to as "The First Jew." As the first person to use his own cognitive abilities to discover and recognize the one G‑d, reject the idolatrous ways of his ancestors and contemporaries, actively publicized the truth of G‑d and was prepared to give his very life for these goals--Abraham epitomized "Jewishness" many centuries before the term came into common use.