The word "Jew" originates from the name of Jacob's son, Judah. The term "Jew" was introduced many generations and decades after him. I wonder whether Adam, Noah, Lot, Abraham and Moses were all Jews?
The term "Jew" is relatively recent. The Hebrew form of it ("Yehudi") first appears (I believe) in Megillas Esther and in Zechariah. Before that, Jews were referred to as "Ivrim" or "(B'nei) Yisrael" (The Children of Israel). (Thanks @IsaacMoses, for bringing up @Menachem's excellent links. I'd like to reemphasize that those should really be checked out, especially the Chabad.org one on the origin of the term "Jew".)
I'll use the word "Jew" hereinafter because it's convenient, but remember that I'm talking about the people, not the name.
Generally, Avraham (Abraham) is considered to be the first of what we now call Jews. We actually have an interesting discussion here on when he became so, and what made that happen.
Traditionally, we do not call Adam, Noah, or Lot Jews. Only from Avraham and down do we start to refer to people as Jewish.
That's not to say that Adam and Noah* weren't holy people, just that they didn't have that same special status that G-d gave to Avraham and his children. But don't worry about them, they've got their own unique places in Jewish history. (Adam, the first man, the archetype of humanity. Noah, the only worthwhile guy in the universe at the time.)
*Noah's relative holiness is disputed by the rabbinic commentators. Some say he was only considered righteous compared to the rest of his (really evil) generation, while others say he was indeed righteous by any standard. I have intentionally left out Lot, as Jewish tradition does not consider him to be much of a holy guy at all, as far as I know.)
The term Jew originally referred to members of the tribe of Judah/יהודה. After the reign of Solomon, the kingdom of Israel split in two. The northern part contained most of the tribes and was thus called the Kingdom of Israel. The southern part contained the tribes of Judah and Benjamin, with descendants of David as king. This became known as the Kingdom of Judah. The citizens of this country were called יהודים, and their language was יהודית.
The Kingdom of Israel was exiled by the Assyrians, and more or less disappeared from history:
Until Adonoy banished Yisroel from His Presence, as He had spoken through all His servants, the prophets; and Yisroel went into exile from its land to Ashur until this day. (Kings II 17:23)
As a result, the remaining Jews were from the Kingdom of Judeah, and would be referred to as Jews in history, even if they were from a different tribe:
Now in Shushan the capital there was a certain Jew, whose name was Mordokhay, the son of Ya᾽ir, the son of Shim῾i, the son of Qish, a Binyaminite; who had been exiled from Yerushalayim with the captivity which had been carried away into exile with Yekhonya king of Yehuda, whom Nevukhadneżżar the king of Bavel had carried away into exile.
Nowadays, Jews refers to the entire people of Israel, colloquially speaking. (Rabbinic sources still use the term Israel over Jew) The term generally includes Abraham, Isaac, Jacob, and all of Jacob's descendants.