I think the Luxembourg Garden near the 6th is what you are looking for. I love that place, and it's been in numerous movies.
Atelier de Lumiere is an amazing show you can go to day or night and stay as long as you want, and not too expensive.
Canal St Martin and Place de Republic - I like this area. There is always something going on at the place, and there are some nice but not fancy restaurants within a few blocks. It's more down to earth but still kind fresh and trendy.
St Paul's is not a place I would recommend ... there is kind of a carousel and some cute shops right near it, but its on a triangular plaza, and it turns out that cars actually drive onto that plaza - in a place that looks like its not for cars. So I would not take a kid there.
Place de Vosgues is in-between St. Pauls and the Bastille. Its ok if you are in the area
The Seine there are very nice walks on both side of the river. The area on the other side of the island from Notre Dame seems a little nicer to me, and there are people hanging out at the river edge day and night. It's a fun environment. Some areas have boats, others have beautiful bridges, others have places to walk past the iconic little magazine stands, where others you can walk down by the river.
Sacre Coeur offers a great view, and is a lively area to walk around. You can also go to the top floor of the department store near l'opera as RS recommends for a cheap thrill.
Les Invalides has a pretty large military museum which might be fun for the kids - knights' armor, Napleonic uniforms, even WWI and WWII stuff. Napoleon's tomb is kinda boring IMHO despite my huge admiration for man himself.
Eiffel - I have been there many times, but last year it was so dissappointing. They broke up the open nature of the park with glass walls. And the walk from there up the hill to the Embarcadero was not only tourists but guys selling cheap trinkets every 10 feet. Plus construction going on. I had such nice memories of there from decades past, and its barely recognizable anymore.
