I’m assuming you mean trad when you say French cuisine. In today’s Paris, French cuisine has quite a spectrum from old-fashioned cuisine bourgeoise that most foreign tourists mistakenly regard as typically parisian to an updated and lighter version “bistronomie”, and to very creative modern French cuisine that is often like “haute cuisine” at bistro prices.
On the borders of the 5th and 6th can sometimes be tourist trap territory, especially around place St Michel and the trashy Huchette quartier (a mini-Fisherman’s Wharf). Not knowing exactly where in the 2-km-long boulevard St Michel you will stay, I’ll suggest Hébé on rue Frédéric Sauton (lovely street) for Med-slanted French, quality of the cuisine, and vibe. If you get tired of French cuisine, Hébé’s younger sibling Ya Bayté on rue Grands Degrés does excellent Lebanese street food, including falafel. Or Café de la Nouvelle Mairie because of the setting on rue Fossés-St-Jacques near the Panthéon and for the continuous hours. In St Germain-des-Prés proper, Huguette on rue Seine comes with enthusiastic approval by all the teens in my extended family, most of their parents, and me… but will most suit if your kids like seafood. For crêpes (a natural with kids), Breizh Café Odéon (continuous hours and 7/7) which has the added bonus of being a 2-minute walk from very good Il Marchese del Gelato ice cream shop… or, smaller and perhaps more difficult with a party of 5, Little Breizh (meal times only) on rue Grégoire-de-Tours.
Near the Arc de Triomphe, my office is in this area and I know it can be a challenge to find restos with a good price/ quality ratio and suitable for families. I’d recommend dinner-only Le Hidé (opens at 6:30pm) on rue Général-Lanrezac for very good trad with a hint of Japanese sensibility (the owner is a French-trained Japanese chef) or, for more hipness that could appeal to the teenagers, Gabylou on rue d’Armaillé. If you want a quicker street-foodish meal, Micka on rue de l’Etoile for excellent Egytian falafel. If worse comes to worst, there is a branch of the family-resto chain Hippopatamus on avenue Wagram @ pl Ternes… steakhouse, quality ok, prices ok, open 7/7 and continuous hours so fine for an early meal (before 7:30) but ambiance is, well, chain restauranty and identikit assembly-line décor… if you were not a family of 5, I wouldn’t recommend it but sometimes these places are a good fit.
Near the Tour Eiffel, either Carette (which is really a pâtisserie-salon de thé but they also do meals) on the place Trocadéro or Les Marches on rue Manutention on the other side of the river next to the Palais de Tokyo for well-priced and very good trad. If you want to give your kids a sample of more contemporary French stylishness/ trendiness, Monsieur Bleu inside the Palais de Tokyo is not bad and has a great terrace with some views of the Tour Eiffel. BTW, the Palais de Tokyo (which includes a free modern art museum as well as artist workshops) is a pretty cool place and popular with the trendier sort of younger Parisians. There is also a very good and large Wed + Sat outdoor food market just outside the Palais de Tokyo on the avenue Président Wilson for a sampling of Parisian life and a respite from all the sightseeing.
For the Tour Montparnasse, either Chez Marcel or La Gentiane, both on the rue Stanislas, for trad French.