Palermo

The Norman Palace - Palatine Chapel or the Cathedral of Palermo would be my recommendations. For the Palace we didn’t purchase our tickets ahead of time last April but we arrived 20 minutes before they opened. Since it is a National museum you have to go through a metal detector which might slow things down if crowded. The Palace has a roof climb which was easy and a crypt you can visit.
The Cathedral has many of the same mosaics as the Palace so if you can only get to one it doesn’t matter.
I would choose one and then just spend some time walking down the pedestrian only Main Street passing the Quattro Canti which Rick references.
If you start at the Quattro Canti the two main streets which intersect leads to all the sights. The Via Vittorio Emanuele is probably your best best. One end is the Norman Palace with the Teatro Massimo (godfather 3 movie famous scene) at the other end. The Cathedral is also along this street.

How much time do you have? Minerals is the Palermo’s top site, a bus or taxi ride up the hill.

We have all day in Palermo. Probably required to be back on ship at 4:30 or 5.

I think Suki meant to say Monreale.
(Darned Autocorrect!)

We also went to Monreale. The mosaics are so similar that we didn’t feel it was necessary to visit both. With such a short time in Palermo I don’t think it’s necessary to visit both. Just walking around Palermo is fun.

Barbara doesn't love mosaics as much as I do. I was very happy to have seen both Monreale and the Norman Palace. However, with only one day you might want to devote some time to the markets, a street-food tour, the archaeological museum, some of the churches and chapels scattered around the city and/or just walking around. It does take a bit of time to get out to Monreale by public bus; a taxi would obviously be faster.