Sicily - Western or Eastern side?

The decision should be based on what you are interested to visit. For example the west offers Segesta, Selinunte, Erice, Zingaro Reserve, lots of wineries, Palermo, etc. The Ionian coast offers Taormina, Siracusa, Mt. Etna, Noto, etc.
In terms of ease of driving or family friendliness, I can’t think of differences. Either side will have places that your family can enjoy. No difference in ease of driving either between the two.

If you choose the eastern side, being closer to Siracusa would be preferable. If you only visit one coast and it is the eastern side, do not stay in Taormina!
You really need to see both sides. How long do you have? Two weeks is a recommended minimum for Sicily.

Thank you so much for the replies! We are looking at one week stay. We likely won't be traveling around different towns too much given that we'll have a senior with us, so more of a low-key holiday with locale that gives us options if we want to pursue some sightseeing. Mostly would be happy to be close to a town that is charming and has restaurants, shops.

All of Sicily will be family friendly so don't worry about that. There is so much to see in Sicily that it is often hard to pick and choose destinations and an itinerary. Hope my comments below give you some food for thought.

If you stay in Stazzo you have the Aci towns to explore, Mt. Etna, wineries, Taormina (lots of uphill walking involved there for your 80 year old). Be aware that the beaches that way are generally rocky. Catania is nearby but since you didn't mention Catania and Palermo I assume maybe you are trying to avoid the bigger cities, but they have much to offer. A daytrip to Catania from Stazzo would be fun. There is a Circumetnea train line that runs from Catania to many of the small towns all around Etna. Might be fun to use that. I agree with Suki's suggestion of Siracusa as an eastern base. If I were to choose the east I would stay in Ortygia and do daytrips from there. Possible trips would include Noto, Ragusa, Modica , Marzamemi, Scicli.

I personally would choose the western side. Palermo has lots to offer and is relatively flat which makes for easy walking. Museums, puppet shows, opera, great outdoor markets, churches, palazzos, architecture, and food! A variety of options that appeal to a range of ages. You can do a street food tour or a cooking class. You can visit Monreale or the beach at Mondello for your younger ones. Segesta to see the temple and ampitheater are nearby. (Again for your 80 year old, there is lots of uphill walking involved there). All ages would enjoy a daytrip to beautiful Cefalu. And by basing in Palermo you can and should avoid renting a car and driving. Cefalù may be another good base. It is such a pretty town, there is a beach for swimming. It is close enough to do a daytrip by train or bus into Palermo or to nearby Castelbuono or Cacamo.

I am not sure if i, personally, would stay in Marsala. There a car may be beneficial. I have been to Marsala but have not spent the night. Marsala puts you close to the Salt pans, wine, nearby Mazzara del Vallo is lovely and super easy to explore on foot. The ruins at Selinunte are spectacular if you have transportation to get there. Your kids will love it there and since the grounds and ruins are so expansive they have a hop on hop off system of golf carts that take you from one area to the other which may help with all that walking in the heat for your older visitors. Give a look at Trapani which puts you relatively close to Segesta, San Vito lo Capo, Scopello and lovely Erice. Castellammare del Golfo would be a low key destination for a laid back oceanside experience and you would be close enough to go into Monreale or Palermo or Trapani to explore.

You asked about ease of driving. With a group of 9 you would need to rent a small bus or van. Are you sure you want to drive? There are lots of issues renting cars in Sicily. You can search the forum or Trip Advisor to see why it is often not recommended. With a group your size it may be cost effective to hire a driver with a small bus. No need to worry about prohibited traffic zones, parking, accidents, etc. Sicily has decent public transportation options also. You could also join a daytrip tour for a day to a given destination and then you avoid doing the driving.

The eastern side has more of the popular tourist sights. The western side has fewer and more laid back. We enjoyed both sides. Marsala was very low key with few tourists and would be a nice place to relax for a week. It’s fairly small and easy to get around. The salt farm a few miles outside of town is nice to visit and nicer than the one at Trapani. Marsala is close enough to Erice, Trapani, Selinunte, and Mazara Del Vallo to make easy day trips.

Driving is easy, but I found the rural roads poorly marked and not always in good repair. The ZTLs (no driving zones) were pretty easy to spot. There are Island wide speed limits and we always drove below the limit. We had a GPS as well as Google maps of Sicily downloaded for offline use. They came in handy for a couple remote sights not in my GPS. We had no problem with renting a car. However, some side streets in some towns are very narrow and you could have issues maneuvering with a large vehicle. In those instances some prior planning on parking locations would be beneficial.

In at least some countries you would need a commercial driver's license to drive the size of vehicle you would need for your group (not to mention everyone's luggage). The usual suggestion in this type of situation is to rent two smaller vehicles. That would avoid the license issue and allow for flexibility in daily activities. Smaller vehicles would also be a lot more maneuverable.