The top tourist sites in Paris are its museums and monuments, but Le Jardin des Plantes is an excellent choice for families looking for a peaceful, educational, and fun destination within walking distance from Notre Dame de Paris. (I suggest taking the Metro, though, if you have little ones in tow.) This garden is France's main botanical garden, and it houses a zoo, a carousel, an amazing animal display (La Grand galerie de l'Evolution), plus several other museums that make up the Muséum national de l'Histoire naturelle.
This was our first time visiting Le Jardin des Plantes; we mainly wanted to visit the zoo (la Ménagerie) because it is cheaper than Paris' newly reopened zoo, Le Parc Zoologique, and focuses on animals at risk of extinction. The Ménagerie is also Europe's second oldest zoo (old enough that many of its animals were eaten during the Prussian siege of Paris in 1870-71!).
We found the zoo interesting, although the many paths around the circular pens make it easy to change direction--we often had to backtrack to see animals we'd mistakenly missed. The foliage also seemed rather blasé compared to that of the Oregon Zoo back in our hometown. The most visited animals might be the red pandas, snow leopards, and monkeys, but of these animals, only the monkeys were awake and visible. I enjoyed seeing the shaggy French donkeys, as well as a sign in front of the raccoons that read "Les Americans en France!" In retrospect, I think the Parc Zoologique (aka the zoo de Vincennes) that I mentioned above is probably a better value for the cost.
However, the rest of the Jardin des Plantes makes it truly special. Check out the carousel that's known as as the Dodo Manège (because it features only rare and extinct animals), then walk past the lovely gardens to the Grand galerie de l'Evolution. This 18th century building's massive appearance hides a strikingly modern hall filled with more than 7,000 preserved animals organized on three levels. Thanks to incredible taxidermy work and low lighting, the gallery is breathtaking. No zoo in the world offers this many animals--from ocean creatures and mammals to insects and birds--for such close viewing. It's truly a stunning tribute to biodiversity, and your family will feel like you've entered Noah's ark rather than a Parisian gallery.
We didn't have time to explore the other museums or gardens at the Jardin des Plantes, but our kids spent a few minutes at the garden's playground before we left via the gate near restaurant la Baleine. This southeast edge of the jardin faces the Grand Mosquée de Paris (Paris' oldest mosque and one of Europe's largest); we didn't pay to tour it but the exterior architecture is beautiful and I've heard the pastries in the tearoom are excellent.
I hope you get to visit this area of Paris someday if you haven't already!