Try the 7th arrondissement of Paris - I stayed there and it was great because it wasn't terribly expensive, yet it was close to many attractions such as: Tour Eiffel, Musee d'Orsay, Musee Rodin, etc.
I do want to mention that the subway system is great - it's like a big spider web and You can get anywhere You want to go with it. It's also fairly inexpensive. I learned the hard way after spending nearly two hundred euros in my first day there on taxis.
If You have lots of time, Pere Lachais Cementary is certainly a place to visit (where the likes of Oscar Wilde and Jim Morrison are buried). It is a little on the outskirts of the city but changing a few subways along the way will get You there.
Good Luck and have an awesome vacation! : )