6 hours to kill at Fiumicino airport?

Grab a taxi, go to your hotel and meet them there.

Unless there is a reason why you need to meet them at the airport, the suggestion to meet at the hotel is a good one.

If you really need to wait for the party who will be getting the rental car because you are going to a hard to reach destination the first night I'd do something like we had to do this summer for a domestic situation:

We took a red eye and arrived in the morning at an east coast airport. Our relative wasn't arriving until 1 pm. We knew we'd be tired and hungry so we just stayed in the airport, had a nice breakfast, read for awhile and took a little nap. A few hours passed quickly and we picked her up and headed to our final destination about 1 1/2 hours away after she landed. We were over our sleepiness and ready to drive.

EDIT: if you have any airline lounge privledges or can pay a day use fee, this might be the time to use your passes.

It sounds like they're all traveling onward from Fiumicino by rental car to a place that's NOT Rome and may not be within easy transport distance so the usual advice to meet "at the hotel" may not apply. I'd probably just hang out at the airport and wait for the rellies to show up.

Where exactly is your grandfather's home town?

Cantalupo nel Sannio in Molise.

Why did everyone assume they were going to a hotel. It was only a two line question. You are just stuck as there is not much you can do. I freshen up, go outside walk around to bit to begin the fighting the jet lag assuming you are coming in on a US flight. The food is decent in the airport. Since you are arriving you may not access to the shopping area but if you can it kills time. You might be tempted to take nap and I would do it.

Perhaps get a day pass for the airline's lounge and take a shower etc.

Given the vagaries of early/late flight arrivals, really hard to plan a specific activity.

Just do not sleep for more than an hour

Completely agree with the airline lounge pass idea. We usually fly over in business class (on miles) and it's so nice to have the convenience of the lounges. Check with whatever airline you're flying to see if they have a lounge at FCO.

I like the airline lounge suggestion, but another alternative would be to walk over to the Terminal 1 arrivals level and spend your time at one of the restaurants there. IMO they're much nicer than the ones at Terminal 3 arrivals level, and the walk is less than 10 minutes.

Suggest you walk over to the Hilton, and hang out there. Assuming you do arrive at 7:30 and have to go through PC, then to baggage claim, it could be somewhere around 9 by the time you exit the secure area. I've done this several times when arriving from the U.S. and waiting for my Italian colleagues to fly in from Milan.
The Hilton has a huge buffet at a breath-taking price along with an a la carte menu, plus, if you ask politely at the desk, you might also get the internet access for the lobby area. You can freshen up in the public bath rooms in the meeting room area if you want. A final plus would be the ease of your relatives' retrieving you at the Hilton rather than fighting for a spot for parking in the arrival area.

Thank you all! Any suggestions on what to do with my luggage. Wondering if our airline would hold it.

Sam

If you want to store your bags after you've collected them, use the storage service at Terminal 3 arrivals.

http://www.adr.it/en/web/aeroporti-di-roma-en-/pax-fco-baggage-services

Otherwise, I would collect your luggage at the carousel and take it with you wherever you decide to go. The airline doesn't want to manage it after your flight is complete, nor would I trust them to do that even if they would.

Are you planning to execute a terrorist attack?
I wouldn't mention to the airport personnel that you have "six hours to kill at the airport", not these days.

Your right!! I have 6 hours to waste at the airport.

I second the buffet at the Hilton. I'm not sure if I've ever seen a breakfast buffet bigger than that! A spectacle in and of itself.