Best Central Rome Neighborhood?

Options will be limited with so little advance notice. This being said, Rome doesn't have an unambiguous "nicest, most convenient area". The most convenient ones are not the nicest and vice-versa.

I'd pick Monti or the Trastevere for a reasonable trade-off. But if it were up to me I'd pick up something at quartieri Salario or on the Coppede area, which are very nice, but not central (need public transportation to reach the main sites).

The campo di Fiori/Navona/Pantheon area is extremely convenient and nice. It is heavily touristed, but that’s the trade off for a central location. I stayed on Piazza Farnese a few years ago and this was a perfect location. Walk everywhere, lovely evenings,quieter than campo di Fiori which gets loud.

I second the recommendation from Mira. The Pantheon neighborhood is convenient, great access to sites, restaurants. If not the Pantheon neighborhood, then I would also check out Trastevere. It has a nightlife.

We had an apartment in Pantheon area. It was the best location for us, we walked everywhere, even to the Vatican and back!

You are close to everything.

I also agree with the suggestions given by Mira--great locations.

I agree with Mira too. We had a VRBO off of the Campo di Fiori and it was very easy to walk or take buses everywhere. Next time I would like to stay in Trastevere as we really liked the atmosphere, so you should look there as well.