I did this last year, because my mileage ticket could only get me to Rome. As said above, you clear immigration in Rome, pick up your checked bags, then go through customs. You then check in for the flight to Palermo, then go through security again. Customs is not the problem, as that's a walk-through. There can be delays at security, but that's not the big problem. The big problems are immigration and baggage claim - they can be fast or slow, and there's simply no way to predict.
So, what I did was buy two tickets FCO to PMO - one 3 hours after my scheduled arrival, and another one 10 hours after. I knew I would be throwing away one of the tickets, and just figured the total cost of the two flights as the cost of my flight from FCO to PMO. In the end, I made the earlier flight with time to spare, but you can't count on that.
And since Iberia doesn't fly from Rome to Palermo, the flight you are looking at is a code-share. Find out the actual operator of the plane, so your daughter will know which terminal and which counter to go to for check in. Particularly since she may be rushing, she can't afford to go to the wrong place.
If she is on a single ticket, it's much easier. She will clear immigration in Rome, but her checked bags will be checked through to Palermo. She may have to change terminals (in which case she will have to go through security) or not, but she will get both boarding passes in the US. So, all she has to do is get to the next flight. Then, she will pick up her luggage and clear customs in Palermo, but as said, customs is merely a matter of walking through the "green channel - nothing to declare" line. Furthermore, if she is all on one ticket and her first flight is delayed enough that she misses the connection, Delta/Alitalia will put her on a later flight at their expense. If she's on separate tickets, it's up to her to figure out how to get from Rome to Palermo, and the cost is born by her.