It's a shortish stint, so sticking to Rome is probably best, but if they want a taste of Tuscan Italy, then a fast train to Florence would be easiest.
In Rome, we stay at 'Holiday Home the Rosary' in the Monti district. It's an active convent, so it's spare, with a midnight to 6am curfew and no TVs in rooms. But nice folks, very affordable, includes breakfast, and Monti has great food and proximity to everything including the Cavour metro station. Pro tip: take earplugs to sleep in case the birds are active at night.
I'd highly recommend the Borghese Gallery for great art, and a nice hike through that garden down to the Spanish Steps with tons of view spots along the walk.
Depending on your preferred activities (sunrises/views, culture & museums, passagiatta & food, or nightlight), there are so many things to do, both at the main sights and in various neighborhoods. We loved the Monti area for delicious local-centric restaurants (2nd breakfast at La Licata nearby, dark chocolate gelato there at Don Nino after spritzers by the fountain, AI Tre Scalini for dinner not far away).
You'll want to make reservations for dinner and maybe for high traffic sites; tourism is picking up again after 3 years of quiet (we were there last month). The first Sunday of the month is Free Museum day, so everywhere will be pretty bustling; pick that day for wandering, hiking, or low-key pursuits. We liked walking through the artsy part of Monti northeast of the convent, and popping into our favorite "secret" park nearby; stairs up from Via Mazzarino lead to the high garden of Villa Aldobrandini. It's laid back, has great views, and locals let their dogs run together up there.
