1. Visit the Louvre on Wednesday or Friday late, after 6 or 7. They are open until 9:45 PM both days. Much, much less crowded at night and a really wonderful experience to look out the windows as the sun sets and the city lights up.
2. Same for the Orsay, Thursday is the late night, until 9:45 PM.
3. Parc Des Buttes Chaumont and Parc de Belleville. Chaumont has beautiful rock outcroppings to climb up, and nice aimless walks along the hills. Belleville has beautiful views of the city.
4. If going to the Marais to eat Israeli/middle eastern food, skip the line at L'As du Fallafel and go to Miznon, 200 feet to the south. Amazing food and great laid-back environment.
5. Holybelly (get there early!) for an amazing breakfast and great coffee. Not really Parisian, admittedly, but a great place.
6. Personally not that impressed by the Carnavalet museum, but was surprisingly intrigued by the underground museum next to Notre Dame, which goes a bit deeper (pun intended!) into the history of the development of Paris and the first settlements on Ile de la Cite.