Fiumicino Airport is a major hub with many long-haul routes and it can often be the case that the best value tickets will include a night change, which is not covered by the airline under any circumstances.
Hotels near the terminal
There are two more luxurious hotels right at the airport with prices starting from 165 eur per night. However, they are within a 3-minute walk of the terminal.
Hotels in Fiumicino
The cheapest hotels are in Fiumicino, about 3 km from the airport.
The vast majority of them only offer chargeable transport (usually around 15 eur), but you can use the Cotral bus service for 1,10 eur, see above in the "Transport" section for more details.
Just beware that the last bus from the airport leaves at 21:00. Prices per night are between 90 eur and 150 eur. A taxi, despite the short distance, comes out to about €20, so always calculate well whether you can get to your hotel by bus or whether it is more worthwhile to stay further away in Rome but with good public transport access.
If you're arriving late in the evening and don't want to pay for a taxi or expensive hotel transfer, consider staying right in Rome, which is always accessible by bus even late at night/early in the morning. However, you will need to allow about an hour for the journey.