Update: this was a great recommendation, thanks Bob! It wasn't the cheapest (110 euros each all in) but my friend and I got two comfortable rooms in a kind of mini apartment, including complimentary drinks in the fridge, and Armando picked us both up separately from the airport, and delivered us back there the next day (only a 5-minute drive). And breakfast was very copious and good -- Roberta kept plying us with food till we had to beg her to stop
Despite being so close to the airport it's in a nice suburban setting featuring a park and birdsong, and only a 10-minute walk from the quay on the Tiber where the small fishing boats tie up and you are spoilt for choice in seafood restaurants. We ate and drank our fill in a cheap and cheerful one for a total of 31 euros ...