I'm to old to be a child but man i have some of the greatest "fluke" luck in the world, it doesn't seem to matter what it is but basically i always seem to squeak it in and everything works out perfect.
So basically after hunting around for various hotels (we were planning on staying in paris for 1 week with 3 adults) or various furnished apartment rentals we absolutely scored in the area/apartment we ended up staying with eve though we started looking for something totally different.
We found the apartment through http://www.parisaddress.com/ though orginally were looking to stay on the "left bank" because thats where it is cool etc but after spending 2 weeks here in 3 different locations i can tell you that you might want to check out the Marais area
Basically the whole apartment for 7 days was only around 1,000 euro so less than $US400 for the week per person.
It's on the 4th floor of a pretty narrow stair case but faces a super quiet interior court yard on a small quiet side street but while still very close to shops, supermarkets,food etc so worked out perfect.
11 Rue Beautrellis, Marais, Paris, France
Basically because of it's location on the Number 1 Metro line (Bastille or St Paul were our closest stops - http://www.aparisguide.com/maps/metro.htm ) it made it supereasy to traverse east /west through paris and then jump on the appropriate north/south metro if and when we eeded to change.