May be French, but the simplest of food: Pancakes!
Yesterday we ate at a relatively new addition to the ethnic restaurant scene, Cafe-Creperie Le Bretagne.
They have two locations, one in the French enclave near Iidabashi, and the other in Omotesando. We went to the first just off Kagurazaka street. It was real, it was authentic, and my daughter ate, even though she has never liked pancakes before. The super thin crepes are made out of buckwheat, and really are like what you can eat in Le Bretagne. They start at about Y700 (i.e. egg/cheese, or egg/ham) and go up to Y2000 (i.e. fine ham/spinach/mushrooms/camembert). One of the common definitions of an ethnic restaurant is that it be cheap (relatively). Well, in this case it was not so cheap. We paid a tab of a bit over Y6000 for two adults and one kid (for 3 dinner crepes, two side salads Y500 each, 3 desert crepes and one cup of coffee), whereas we usually get out for about Y5000 and less at the various other places we frequent. Also there were no leftovers to bring home. However, it made a nice change! The menu is French/Japanese, but if you ask they also have one in English/French/Japanese.
Website: http://www.le-bretagne.com (only in Japanese at this time!)
Email: info@le-bretagne.com

Omotesando: 4-9-8 Jingumae, Shibuya-ku, Tokyo; 03-3478-7855 Hours: 10:00-22:00
Iidabashi: 4-2 Kagurazaka, Chiyoda-ku, Tokyo; 03-3235-3001 closed Mondays, 11:30-21:00

PS -- here is a nice cake to make before you go out. Then you can eat desert at home! (Vanilla sugar is a baking product used in Europe. You can substitute liquid vanilla.)
The most simply Breton cake: le Far Breton
ingredients: 250g of flour, 150g of sugar, a packet of vanilla sugar, 4 big eggs, 1 litre of milk, 100g of raisins, a little glass of rum
Mix the flour, the sugar and eggs one by one. Add the vanilla sugar and the milk little by little. Pour the dough in a dish covered with butter and bake in a warm oven between 45 minutes to 1 hour. It is possible to replace the raisins by prunes soaked in rum. It is also possible to drink the glass of rum. The true far has neither raisins nor prunes.

PPS -- Yes, I know Breton is in Nova Scotia, Canada, but there are historical reasons for including this recipe here!

Cafe-Creperie Le Bretagne is definitely worth a try. We took my mother there (she is from Bretagne, France) and she loved it; apparentely it reminded her of the crepes her mother used to make by the dozen. She was also thrilled to be able to speak breton with one of the waiters.

I can also recommend Le Bretagne (both Tokyo branches) - we've been going to the one on Kagurazaka since it opened in 1996. Other child-friendly qualifications apart from the menu (which they will adjust to your child's picky taste, ie our son often wants just a cheese and tomato pancake, without the ham/mushrooms etc on the menu, and they are happy to oblige) include a big booster cushion to help smaller children reach the table, though no actual high chairs. The owner has three young daughters who are often around the Kagurazaka branch, and the staff are very welcoming to kids.

Weekends are more expensive, but weekdays I think they do a cheaper lunch menu.

I can also recommand Le Bretagne, although it's not cheap.
They have take-away branches in Iidabashi and in the food court at Carrefour Makuhari, as well as restaurants in Sapporo and Osaka I think.