Dollars is very low compared to euro, so you will have to save a lot before going.
For plane, save about 1000 and it should be more than enough.
For hotel, I would recommend hostels, "auberge de jeunesse" - there are plenty in many, many towns, and some have individual bedrooms. You'll have to buy their card, and then it will be very cheap to sleep there; they usually offer free breakfast, and a kitchen where you can cook.
If your family wants to come with you, consider home swaping (you offen have to suscribe to a club, like intervac), which is very cheap and practical.
Where do you want to go in France ? Only Paris ? Paris + Riviera ? Paris + Riviera + Châteaux de la Loire ? Mont-Saint Michel ? French Britanny ? Strasbourg ? Toulouse ? Dijon ? It will change a lot your budget.