Our cruise ship docks for a day (8a - 5p) in Catania on a Saturday in October. We like to wander, take photos, check out the local spots, and see new things. Should we stay in Catania for the day, or make a day trip to either Siracusa or Taormina? We have not been to any of these locations. thank you!

I'd pick Taormina over Siracusa and Catania.
Taormina has a very beautiful "April 9th" square (Piazza IX Aprile), a spectacular Greek theater, and plenty of shops and restaurants to people watch and enjoy your outing.
If there are singers singing on April 9th square, then find a lunch spot near by to enjoy the music.

Taormina !

Siracusa but really Ortigia which is the ancient section of Siracusa. You have all of the great ruins in Siracusa(the big theatre and caves) and then the magic of Ortigia. At one time,Ortigia outside of Athens was the biggest city in classical times.The harbor alone has over 4000 ships at its bottom from very famous battles through the years.Archimedes lived in Ortigia and developed the process of using a mirror to blind the ships by deflecting the sunlight into their eyes!(Can you tell I was a history major?)
Ortigia is a 1 mile by 1 and half mile island connected by a bridge to the mainland portion of Siracusa with wonderful restaurants,shopping and gorgeous seaviews enjoyed by walking the sea wall around the island.It is the only place outside of Egypt that papyrus grows(great shops sell beautiful pictures) and has recently been discovered but still retains the old world charm and culture.
Taormina has a beautful Greek restaurant and every cruise ship person docking on the Eastern side of the island! It has tons of touristy restaurants with high prices/picture menus along with all of the "Made in China" tacky souvenir shops.Nice to visit once but never again as it was so over run with people that you could hardly walk down the main street in town.
I have traveled extensively around Sicily and have been to Ortigia many times as my daughter lived there for 5 months and if there is a heaven on earth other than Cape Town South Afrtica-its Ortigia! Just saying...

Make that Taormina has a beautiful Greek theatre-sorry!

For an 8-5 stop I strongly recommend Taormina. Yes, Ortigia is interesting but not for the time you have. The Greek theater in Taormina is stunning, more so than the ruins in Siracusa. Taormina is great for a short stop and that's how much time you have.

I'd also pick Taormina. We liked it the best by far out of the 3 places you list. If you find yourselves with spare time, you could go up to Castelmola, a fairly steep uphill walk with fabulous views at the top.

Kay