It's hard to be helpful with such an open ended question, a lot depends on the interests of your family and the ages of your children.
Our kids all liked the underground tour but they were all teenagers.
You are spoiled for choices. Three castles, Palazzo Reale, strolling along the lungomare, several underground ruins in the city, seeing the sights and sounds in the small shops of the historical center (see the Christmas displays on San Gregorio Armeno and pop into the small "doll hospital" if that is of interest). We took a pizza making class with Eat in Italy food tours - not cheap, but a lot of fun.
For "off the beaten path," Pozzuoli right outside the city has a wealth of archeological sites that are not visited by many. Though transportation could be a challenge. I recommend the Flavian Amphitheater there, which I believe is easily accessible by public transportation. It is a smaller than the Colosseum in Rome, although unlike Rome, you have free access to wander the underground area at your leisure amongst the ruins. We had a blast there.
See www.napoliunplugged.com for more ideas.Edited: 09 May 2018, 23:12