The taxi fares from the city center to FCO is 48 euro (all inclusive flat fee imposed by the City of Rome).
Early on a Sat. morning, the taxi ride from anywhere in the city center is 30-35 min max.
The train fare on the Leonardo Express train (32 min ride from Termini station) is 15 euro per person (one way). Since there are 3 of you, then the expense would be similar by taxi or train (48 euro vs. 45 euro).
Both are reliable means of transportation. The train is not affected by traffic, but at that time in the morning on a Saturday, there will be no street traffic either, so the taxi will be just as fast.
I would stay anywhere you care to stay in the city center, then take a taxi in the morning for the amount I mentioned. If you absolutely want to go by train, then to save money you should stay near the Termini station, otherwise you have to pay transportation to the station and then the train fare.
The airport hotel is an option. There is the Hilton right at the airport where you can walk to the terminal. Not cheap though.
Alternatively you could stay in the town of Fiumicino, next to the airport. There are cheaper hotels there and the 10-15 min taxi ride won't cost much. Obviously Fiumicino is not Rome, but there are restaurants you can go to, and in summer you can go to the beach.
