The centro storico
The default, and worth the money on a short trip. The historic centre is not one place. The Pantheon and Piazza Navona sit in a dense knot of streets that is quiet surprisingly early; Campo de’ Fiori, ten minutes away, is a market by day and a drinking square by night. Booking “the centro storico” without knowing which of those you are next to is the single most common way to be disappointed by an expensive room.
What you get is the thing the area is famous for: you finish dinner and walk home past the Pantheon rather than working out a bus. What you give up is a metro station, a lift in many buildings, and roughly a third of your accommodation budget. Rooms here are old, which is charming and also means small, stairs, and windows onto stone that carries sound.
Choose it if the trip is short and sightseeing-led, and if you would rather pay for location than for space. Skip it if you are travelling with heavy luggage, need a lift, or want a quiet room without paying for a courtyard-facing one.