How early do I need to book restaurants in Rome?
Lat year, I made reservation for only one restaurant (Metamorfosi, and it was not even full!!) in advance (2, 3 weeks by email).
All other places, I went ahead and reserved on site for either the same day/evening or for later in my vacations (Pierluigi, Ditirambo, Roscioli ) or walk-in (Bir & Fud, Tonino's)
I would at least check the best way (AND TIME) to make reservation for the restaurants you wish to go to, (telephone, web, email, ... ) and make reservation 2, 3 weeks before you leave.
If you speak italian, it will be much easier, especially if you carry a phone with you to be able to call a restaurant during the day to ask for a table for the evening.
Disclaimer, I was alone, and did not really care if I missed out on the "best".
It really does depend where and what night. For instance, Taverna dei Fori Imperiali is quite small, and books up at least 4 days in advance. But most places you'll be ok if you reserve the day before. Flavio Velavevodetto, Perilli and Trattoria Monti all get full on weekends.
Having been to Rome 4 or 5 times, and going for a 5th or 6th time in mid October as well, we don't plan on booking until a couple of weeks before we go for the 1 or 2 "musts" on our list. The rest we'll just leave to booking a day or two before, which always seems to work out well for us.