The way I see it the first night of Hanuka ought to be Thursday night [the first day should be December 7]. This would be based on the idea that the actual beginning of the Jewish month should be at the time of the molad [when the path of the moon and sun are on the same longitude.]. It is not that I am truing to disagree with the rambam that says the present day Hebrew calendar is from Mount Sinai. It is just that his position seems to me to be untenable. I have no fun out of disagreeing with the Rambam but in this case I have to. The present day calendar was the calendar that was accepted in ancient Athens and instituted by Meton. It is not mentioned in the Talmud anywhere. The idea that Hillel the second sanctified it is also simply nonsense. If he had done so it would have been mentioned in the Talmud.
lacking a specific calendar we have to fall back on the simple  idea of the Torah itself and the Talmud in Sanhedrin that the time of the beginning of the month is the conjugation. [see the opinion of Rava and Rav Ashi in Sanhedrin as rashi explains them there.]