Agree completely with Zoe. If budget will allow, I'd definitely hire a private car. She's a bit elderly to fly into Rome, change airports with luggage and then take the flight directly to Comiso - and on top of all of that a bus!?. Ragusa isn't an easy place to get to for a non driving individual. Factor in the age/exhaustion issue. Even though I would expect the taxi/private driver to be pretty expensive, it should certainly not be expensive enough to justify a $5k airfare you're looking at to get her to Comiso!
A good ballpark figure for JFK/Catania is probably + - $1600. Even if the taxi is several hundred eu, it would be worth the money.
If she flies Alitalia round trip to Catania, she won't have to change airports, or even get her luggage in Rome. They should be able to offer her special assistance through the Rome airport. I'd "google" private drivers at Catania airport and see what it would cost to get her directly to her friend's home in Ragusa. TA may have some recommendations for a RELIABLE service.
At her age, convenience should be a high priority. I wouldn't even "mess" with Ryan Air. Stick with whatever you can get on Alitalia. Much better service and generous luggage allowance and more professional overall in my experience.