If you go by the actual molad [conjunction of Sun and Moon] the Tues night should be the start of Hanuka.

But the calendar everyone else goes by is usually different because they are basing it on the average new moon. However it seems to me that the date of should be by the actual time of the molad, and that ought to count as the first day of the month.

The basic reason I say this is because of the Gemara in Sanhedrin page 10 where the time for the new moon does not depend on the court of law establishing it, but rather on the right time. As R. Elazar puts it: "If the court sanctifies it in the right time, then fine; if not, then the higher court in heaven establishes it."

So what matters is the right time. Is that when it can be seen, or when the actual molad is in fact? That seem to be a debate between the Gemara in Sanhedrin and the Gemara in Rosh Hashana [I seem to recall that is on page 19.] 

So why choose one Gemara over the other? Well as David Bronson told me once that what makes my idea interesting is the fact that there is no court of law to in fact establish the date. And I would have to add the fact that the calendar that everyone is going by is not mentioned in the Gemara. [If Hillel II in fact established the calendar you would imagine that somewhere in the Gemara someone should have mentioned it. So we have to say it got to be the custom to use it during the time of the geonim. [There was readily available a calendar in use at the time that got the solar year and lunar year to correspond more or less that had been in use for about a thousand years from the time of Meton in Athens.]