Personally I'd just book a night in the relatively new airport Novotel. Rates are reasonable if you book ahead. Don't book breakfast as you won't really have time. You might need to use a taxi (NT$150?? US$5) in the morning if the shuttle bus hasn't started when you want to leave or is full.

(It's the only complaint I have about the Novotel - the shuttle bus isn't really regular enough, doesn't start earlier enough to catch the first flights and is sometimes full.)

I wouldn't consider a trip into Taipei unless you have friends there.

I would only consider staying overnight in the airport if you have got lounge access for food/showers. Having said that, most won't open in time for you to have much time there before going to the gate.