BZZZZZT! We're sorry, but Herod was an Edomite (descendent of Esau), not a Jew (descendent of Jacob).
Not quite. Herod's father was an Edomite. But the Hasmoneans forcibly converted the Edomites to Judaism in about 125 BCE, roughly a generation before Herod's father was born. From the writings of Josephus, it seems like the forcible conversion stuck, and the Edomites became believing Jews. So, on his father's side, I think it would be fair to say Herod was Jewish.
Herod's mother, on the other hand, was a Nabatean, not an Israelite or an Edomite.