What is the best hotel convenient for a one night stay close to the Rome airport?
The airport Hilton is connected to the terminal
If price not a factor, then the Airport Hilton. If you encounter sticker shock, then check out the Hilton Garden Inn, which is reached via a shuttle bus.
Some like to stay in nearby Fiumicino:
https://search.ricksteves.com/?button=&date_range=1y&filter=Travel+Forum&query=Fiumicino+hotel
Just be aware of the high taxi fares to/from Fiumicino.
Welcome to the forum.
Google Maps or booking.com will show you which hotels are close to the airport.
If you’d like to share your dates and your budget, I can google it for you and come up with some options.