Well, I think you are in quite a predicament! Being without a car if you wish to leave San Diego will be quite challenging. Amtrak does offer some service between San Diego and Los Angeles through Orange County... but if you wish to go anywhere other than the station you will pretty much need a car, so the same problem exists.
Within San Diego, I think there is plenty to fill three days time. The holiday season means that a lot may be closing up early or opening late... but the nightlife should definitely still be around as the New Year draws closer.
As for Mexico... I think most Americans would say avoid TJ at all cost. It is not as dangerous as it used to be, granted... but if you are of legal US drinking age (21), then there is really not much to be done there... bar cheap medicine and gambling. If you wish to go further into the country... I think you would have a great time as Mexico truly has some breathtaking scenery... but the time limits and lack of transport will haunt you twice as badly South of the border as north.
So... in short. I would say stay in San Diego. If you change your mind an rent a car, drive out for the day to some beaches... but I doubt they would be too crowded... or take the 2 hour or so drive to LA and hit the highlights. (Are there any??)
Good luck, safe travels, and welcome in advance to the state of the "Governator"!