A "Beit Chanin" is mentioned in the gemara in Pesachim 57a:
"With regard to the prominent priests and those like them, Abba Shaul ben Batnit said in the name of Abba Yosef ben Ḥanin: Woe is me due to the High Priests of the house of Baitos, woe is me due to their clubs. Woe is me due to the High Priests of the house of Ḥanin; woe is me due to their whispers and the rumors they spread. Woe is me due to the High Priests of the house of Katros; woe is me due to their pens that they use to write lies. Woe is me due to the servants of the High Priests of the house of Yishmael ben Piakhi; woe is me due to their fists. The power of these households stemmed from the fact that the fathers were High Priests, and their sons were the Temple treasurers, and their sons-in-law were Temple overseers [amarkalin]. And their servants strike the people with clubs, and otherwise act inappropriately."
The Tosefta in Menachot 13:4 brings a slightly different version, in which Beit Chanin is Beit Elchanan and Beit Piakhi is Beit Phiabi.
Abba Shaul ben Botnit lived shortly before the destruction of the Temple. According to Wikipedia, it's possible that Beit Chanin is Beit Chanan (Ananus ben Ananus in Hebrew is Chanan ben Chanan (חנן בן חנן)). One possible reason, other than the similarity between Chanin and Chanan, is that this idea is brought by Abba Shaul in the name of Abba Yosef ben Chanin, who at times was also called ben Chanan - perhaps he was of the Chanin/Chanan family and lamented the actions of his relatives.
Rabbi Shaul Lieberman in his book Greek and Hellenism in Jewish Palestine, pg. 138-140 in the Hebrew edition, wrote that he believes that the midrash in Shir Hashirim Zuta 5:14 (NY MS) (the Parma MS of the midrash can be found here, but it's missing a key phrase, hence the need for the NY MS) refers to Chanan ben Chanan. The midrash says (my translation):
"...another explanation, "hurry my beloved", in the days of Menachem and Hillel, that a dispute rose between them, and Menachem left, him and eight hundred students with him, dressed in golden bastriki, and Chanin ben Matron came, and kicked [him] ben Yehudah, brother of Menachem, and killed him. And Elazar and his students with him rose up and [he] slew Elchanan and they hacked him to pieces..."
According to Rabbi Lieberman, the death of Chanin refers to the death of Chananyah the High Priest and the death of Elchanan refers to the death of Chanan ben Chanan. The "students" here were the Edomite followers of Elazar ben Chananyah the High Priest and commander of Edom during the Revolt.
According to Josephus, Chanan was one of the people killed by a mob of Edomites.1
This ties in well with the version of the Tosefta of the first story, that refers to Beit Chanin/Chanan as Beit Elchanan.
1 In his book "Eretz Yehudah", pg. 251-253, Rabbi Shmuel Klein made a compelling argument why these "Edomites" (or "Idumeans") were really southern Jews (בני דרומא). He compares, for example, Sifrei Zuta on Chukat (Genizah MS) which says: "היו תלמידים אדומים לבית שמי..." (there were red/Edomite students in Beit Shammai).