This topic has been on my mind lately. According to Josephus, Herod's father Antipater was the son of Antipas, one of the Edomites converted by Yochanan Hyrcanus. I asked one of the rabbis in my yeshiva and he said that though in the first place the conversion was problematic, בדיעבד it counts, which means that Herod's paternal side was Jewish. The problem starts with his mother Cypros, who, again according to Josephus, was born a Nabatean-Arab princess (Antiquities 14:7:3). There is no evidence that she properly converted, though as the note there in the Gutenberg version states, her name as it appears in Josephus is the Judaic/Hebraic version of the Greek name Cypris.1
Though I can't find it right now, I remember that the Talmud concludes that he was a "Canaanite slave", though I'm not really sure how that makes sense in light of Josephus - was that statement meant to be taken at face value? Yes, I'm aware that there are discrepancies between Josephus and the sages. But is that really the case here?
Therefore, I was wondering whether anyone was aware of sources that discuss the status of Herod's Jewishness, other than the gemara. I'm particularly interested in sources that are familiar with what Josephus or other historians wrote about his lineage.
1 Likewise, many Greek names were altered through the Second Temple and Talmudic eras' Hebrew/Jewish Aramaic, such as Nicodemus becoming Nakdimon.