Nicole Czarnecki's answer to "If my maternal grandmother was Jewish, but my maternal grandfather wasn't, am I Jewish?" on Quora

Yes. Here’s the little secret that few like to tell: the one-sixteenth rule comes right from Tanakh (Specifically, Divrei HaYamim Alef). Because Anti-Semitic and other racists perverted it, what was a protection of bloodlines for as many generations as possible—i.e., that you could be considered a given ethnicity so long as you were one-sixteenth that—became perverted into the one-drop rule.

14 The sons of Manasseh: Asriel, whom his wife bore--his concubine the Aramitess bore Machir the father of Gilead;

15 and Machir took a wife of Huppim and Shuppim, whose sister's name was Maacah--and the name of the second was Zelophehad; and Zelophehad had daughters.

16 And Maacah the wife of Machir bore a son, and she called his name Peresh; and the name of his brother was Sheresh; and his sons were Ulam and Rekem.

17 And the sons of Ulam: Bedan. These were the sons of Gilead the son of Machir, the son of Manasseh.

18 And his sister Hammolecheth bore Ish-hod, and Abiezer, and Mahlah.

19 And the sons of Shemida were Ahian, and Shechem, and Likhi, and Aniam. {P}

[JPS 1917, via MechonMamre.org]

Machir ben Menashe was one-sixteenth Jewish and established the cutoff when he married a daughter of ‘Am Yisra’el. Why? Because Yitzchak, through whom B’nei Avraham are counted, was the first Jew (and Jews are called thus partly because Yehudah—Judah—is the tribe through whom is the ancestral line of Mashiach—assuming that there is a literal Mashiach and regardless of whom one believes him to be).