I stayed near the train station, too, which was ideal for me, but I wouldn't recommend it for you. I think Rick suggests the area off the Promenade des Anglais a bit west of the Old Town, commenting that the infrastructure of the hotels there makes for a more reliable experience. (I am paraphrasing very inaccurately, I'm sure, but I think he meant that very old buildings sometimes mean less-reliable elevators, plumbing, etc.)
There is an endless variety of side-trips available from Nice. I didn't go to Monaco myself (which tells you how I prioritized it), but it does have a well-regarded aquarium, a botanical garden, and the Prince's car collection. I suspect it may have more of interest to your children than most other places along the Riviera, whose claims to fame lean heavily toward picturesque architecture, views, art museums and gardens.
Rick has a book focused on southern France that is said to have more complete coverage of that area than his general "France" book (which is what I used). There is so much to do in that area that you really, really need a book like that to help you focus on what will be of greatest interest to your family.
Provencal markets are fun for an hour or so. The Nice tourist office (one branch on plaza right outside the main train station) should have information about those in the immediate area. That office was extremely helpful to me last year. It was able to provide maps of some of the surrounding towns I was headed to.
I don't have children and am not very good at this, but these activities might appeal:
I'm pretty sure I saw folks biking along the Promenade des Anglais in Nice. I'm sure there are bike rentals in some of the other coastal towns as well. You can probably find that information by Googling. Nice is large enough that there may be bike tours on offer.
You can walk part or all of the way around several of the nearby peninsulas, watching the Mediterranean as it crashes on the rocks. The path will not always be paved, so you really need lace-up shoes, and you should skip this if anyone is dealing with a sprained ankle (or turn back at the first sign of difficulty). I walked on Cap Ferrat and Cap d'Antibes. I thought the walking was a bit messier on Cap d'Antibes. Maps from the T.O. in Nice.
Villefranche-sur-Mer has a citadel with several small museums inside, at least some of which were free. The kids might enjoy seeing the citadel itself even if the museums don't appeal, and it won't take the adults long to visit the museums.
If you're interested in art museums or gardens, post again for suggestions. Or just throw a dart at a map.
What time of year is your trip? If you're thinking "beach", you'll need to do some research. Many European beaches are uncomfortably rocky rather than sandy. And I think a lot of the beaches along the Riviera may be private.