Her parents also say they might be descended from the Vilna Gaon

The things you learn as you go along.  From Contra Smith On Jewish Selective Immigration by Scott Alexander.  

This isn't the way most American Jews remember their own history; family lore usually focuses on how our ancestors were the poorest of the poor. My great-great-grandfather was a chicken farmer in Poland. He first emigrated to Germany, but felt like the German Jews were too stuck up and contemptuous of poor Polish Jews like himself, so he booked passage to America. I asked my Jewish housemate, whose family has millions of dollars and all went to Ivy League schools; she says her emigrant ancestors were "a Kosher butcher in Minsk and some guy who floated logs down the Dneiper River".

(her parents also say they might be descended from the Vilna Gaon, but every Jew says their family might be descended from the Vilna Gaon)

 I have never heard of Vilna Gaon.  From Wikipedia.

"The sage, our teacher, Elijah") or Elijah Ben Solomon Zalman (Sialiec, April 23, 1720 – Vilnius October 9, 1797), was a Talmudist, halakhist, kabbalist, and the foremost leader of misnagdic (non-hasidic) Jewry of the past few centuries.  He is commonly referred to in Hebrew as ha-Gaon he-Chasid mi-Vilna, "the pious genius from Vilnius".

Through his annotations and emendations of Talmudic and other texts, he became one of the most familiar and influential figures in the rabbinic study since the Middle Ages, counted by many among the sages known as the Acharonim, and ranked by some with the even more revered Rishonim of the Middle Ages. Large groups of people, including many yeshivas, uphold the set of Jewish customs and rites (minhag), the "minhag ha-Gra", which is named for him, and which is also considered by many to be the prevailing Ashkenazi minhag in Jerusalem.

Noted.

So Ashkenazi Jews claiming suspected descent from Vilna Gaon seems roughly the equivalent of American families claiming suspected descent from Native Americans.  

Interesting the things you don't know which are widely known within their own communities.