Sicily where to spend "extra" day?

You could add that extra day to Trapani, giving you time to take two great day trips:

Erice — (drive or take the funicular) — spectacular views

Favignana(by ferry) — rent bikes and explore this pretty little island

We spend 2 weeks in Sicily in May and loved it; we toured the entire island and hit all the stops you mention.
Taormina would be my suggestion for your extra day. Taormina offers the amazing ancient Greek theatre, Ionian sea and beaches, bustling main streets offering unique shopping and wonderful food and local wine tastings. It looks from your note that you are planning to visit Mt. Etna which is close to both Taormina. We booked a guided tour up Mt. Etna and it was really informative and exciting.

Erice. The views are unbelievable.

At first read I was going to advise stopping at Parco Archeologico di Segesta on the way to Erice and spending the night. However, on second thought you can probably stop at least one of these on your way to Trapani.
I think the best use of a extra day would be in Ragusa. There are so many towns to day trip to.
Good luck and happy travels.
Gina

We spent two nights in Ragusa and really wish we had a third. We had one day for day trips and really would have liked more time.

There are lots of worthwhile places that could use that night, but what stands out to me is that you just have 3 nights for Palermo, which has a lot of sights. That seem short, especially if you will be arriving in Palermo jetlagged.

Ortigia is charming and I don't exactly think 4 nights is too many, but one reason to have sufficient time there is to see some of the Baroque villages, of which Ragusa is one. You have a total of 6 nights in that area and only 3 in Palermo.

We visited Villa Romana del Casale while driving from the Ortigia area to Agrigento. I’m not sure a night is needed there. While there are some nice views from Erice, there’s little to do there in the evening after the day trippers leave and the shops close about 5pm.

The ruins in Agrigento take 3-4 hours to see, so depending when you get into town you might be rushing it. We spent two nights there. The first was just wandering from our B&B to a ruin not in the main part of the park. On the way back we wandered into Giardino della Kolymbethra. On our full day there we visited the temples. Maybe add the night there.

Another consideration for the extra day could be Cefalù. For your Etna day, ensure you have a plan B in case it is erupting when you’re there.

Another vote for an extra day in Palermo, especially as you could take an entire day to visit Monreale and the cathedral and cloisters.