Avraham Avinu is often called "the first Jew".
When did he become Jewish. Was it when:
He recognized G-d?
G-d spoke to him for the first time?
Covenant Between The Parts?
When G-d changed his name?
When he had a circumcision?
When Yitzchok was born?
When he passed the 10th and final test?
Some other event in his life?
Or perhaps he was never Jewish and we just mean that he was the first in what would eventually become the Jewish nation.
"Jewish," as @Yishai points out in the comments, is an anachronism. I don't think we're talking about being what we'd call Jewish nowadays - being a member of the Jewish Nation. It's pretty irrelevant to ask whether you'd marry off your daughter to Avraham Avinu or whether you'd count him for a minyan, since neither your daughter nor the other nine guys could have co-existed with him.
I think the sense in which we call him the "first Jew" is that he entered into a covenant with God that would designate his descendants as the "Chosen People" (whatever that means). The first place we have this in the Torah is at the Covenant between the Parts (Gen. 15). Therein, God foretells what would be the forging process of the Jewish Nation through exile and redemption and promises that the conclusion wold be the Jews' acquisition of the Land of then-Canaan. I think it's fair to say that at that point, then-Avram was designated as the progenitor of the "Chosen People."
Perhaps it was when he rejected idolatry. Hhaza"l explains (exact source I don't know immediately) the reason Mordekhai was called a Jew, Yahudhi, although he belonged to the tribe of Binyomin, is because all who reject idolatry are called Jews, Yahudhim, (as if there is a broader and a more exact definition for the term). This is based in the fact that the name Yahudho derives from the 4 letter Name of HaShem, י and ה followed by ו and ה, combined with the phrase הודיה (thanks/praise) and/or any of its variants; thus the name means roughly "one who praises/thanks HaShem", as opposed to idols.
As to why all of the people of Israel are referred to as "Jews" can certainly be explained in this way, and also that the tribe of Yahudho was the primary surviving tribe during the Babylonian exile, after many of the northern tribes of Israel were majorly exiled by Assyria beforehand. Albeit, amounts of all tribes do exist within the Jewish people today. And in the future, Moshiahh will be able to say who is who.
This article from Chabad.org, Was Abraham Jewish? On the Identity of the Pre-Sinai Hebrews, pretty thoroughly discusses this topic.
I'm not sure how to elaborate on it without rewriting the whole article, but after discussing the different times Avraham was chosen by G-d and the Covenants made between G-d and Avraham, one paragraph says:
Accordingly, when we refer to Abraham as the first Jew or convert, it does not mean that he was actually Jewish in the sense that we know today—in the sense of a binding obligation. Rather, he had the technical status of a Noahide, just as any other person of the time (albeit one that was given additional unique commandments such as circumcision, in which he was indeed obligated). It was not until his descendants stood at Mount Sinai and G‑d proclaimed, “You shall be to Me a treasure out of all peoples . . . and you shall be to Me a kingdom of princes and a holy nation,” that we became the Jewish People.
Please read the article for context.
Only after Sinai could a person be considered a Halachik, or legal "Jew". From Sinai and onward if your mother is Halachikly Jewish, then you are Jewish. Another way to become a Halachik Jew after Sinai is through a legal process of conversion.
Since Avraham did not live during or after Sinai he would not be considered a Halachik Jew. He would be considered a Ben Noach.
He is our forefather and his monotheistic philosophy remains the bedrock of Judaism. We continue his mission in the world to make known the correct idea of Hashem. However, from a legal standpoint he would not be considered a Halachik "Jew".
I would argue that Avhraham never became Jewish, and he is only the first of what we would later call Jewish.
I give three reasons for this.
Avraham's children do not all inheret from him. That is, many of his children are listed as being from a different nation than that which the "Jews" would be called. (i.e. Ishmael Zimran, Jokshan, Medan, Midian, Ishbak, and Shuah.)
If one wanted to argue that Judaism started in the time of Avraham, then it would seem that status of 'first' really belongs to Sarah, for in Halacha you are Jewish based on the status of your Mother. It appears that since only Yitzchak was born to Sarah, that is the only family line that becomes Jewish. (However this argument fails with Yitzchak and Rivkah, whee Eisav also breaks into other nations and is not considered Jewish) It is only after Yaakov, when we are called Bnei Yisorel where we remain a single nation from then until now.
Judaism, Christianity, and Islam are called the "Abrahamic faiths", this would imply that Abraham was not exclusively Jewish by common consensus.
As an aside: The term "Jewish" or "Jew" refers back to the split between the 10 tribes which became "Israel" and the Yehuda and Benjamin, which became a single nation of Yehuda. I would suggest that the honour of "First(recorded) Jew" goes to Mordechai. In the Megillah Ester, Mordechai is labeled a "Jew".. where he is called a yehudi, even though he is listed as from the tribe of Benjamin.
I found this article on Avakesh.com that brings many different opinions about when Avraham Avinu became Jewish, if at all.
Some of the sources he brings are a little less than ideal (e.g. children's books and wiki answers), while others are pretty good.
