Overnight outside of Rome
 

We are landing in Rome and need a spot to stay overnight outside of the city after a long flight. We will be picking up a rental car at Fumiciano and heading towards Peschici in the Gargano on the east coast. Something an hour or two outside of Rome in that direction would be ideal.
We are a family of five - 3 kids ages 5,10,12.
Something easy to drive to, not in a major city. Smaller town preferred so we can have a walk and dinner our first night,
Budget approx. 250 Euro.
A family-style room that sleeps us all would be ideal.

Even a suggestion of a town and not a specific place to stay is a help. I don't know where to begin choosing a spot. A favourite small inn or B&B sugggestion would be greatly appreciated!

Tivoli is right on your route. We once stayed at an inn called the Villa Adriana, right outside the archaeological site of the same name. It's the summer villa of the Roman Emperor Hadrian, and would be worth a visit before you leave. The inn also had what looked like a good restaurant, but we had already eaten when we got there. It was in a secluded and quiet spot. However, this was about 20 years ago, so I don't feel that I can recommend the inn. However, the town is nice, and has two other villas people visit. The most famous is the Villa d'Este, which has a fascinating water garden.

Tivoli is a little less than an hour's drive from Fiumicino airport.

Depending on what time of day you arrive and what season you are traveling, you might consider picking up the car and heading straight for the beach. Either Ostia Lido or Fregene. You will certainly find family accommodations there. I do see the sense in wanting to go east, and be on your way to Peschici, but if you sort of blanche at the prospect of traffic jams, and if you are arriving early enough to enjoy the beach for your walk, might strike some people as the right antidote.

It will mean a longer drive the next day, but just a thought.

In case it wasn't clear, the beaches of Ostia Lido and Fregene are on the Mediterrean, right at the edge of the airport, about a 10 or 20 minute drive from the car rental office.

Our flight lands at 10:45am so we want to get a bit of driving time in that day. It will be the end of June so the weather should be nice. We thought of going south to the beach but really want to stay along our eastern route towards Peschici.

I think Tivoli would be your best choice on that route. If you go any farther, you'll be in a rather sparsely populated area, where the places on the map are little more than hamlets.

The beaches around Fiumicino (Ostia, Anzio, etc.) are so close to the airport that a visit there would hardly be a detour.
But there are many hotels along both itineraries
via Tivoli - Avezzano - Sulmona
or
via Frosinone - Monte Cassino - Isernia.
A stay at Lake or Frascatgi would be possible too.

It should be: A stay at Lake Albano or Frascati would be possible too. Sorry!