With Difficulty!
I spent a long time trying to find a great deal! Unfortunately, there are not many offers to be had. They are a sly bunch!
I can recommend the Davy Crockett ranch option which have 2 bed cabins that were fab, they can sleep 6 (2 in the living room). They are a short trip from the park and have an active village centre which has a shop, bar, kids activities and a restaurant. This is the cheapest accomodation which is 'part of the magic' However, you could try the satellite hotels which Disney recommend and are close to the action but you have to make your own way there i.e. via train but they are just boxes with rooms and dont have those Disney touches.
I dont think you will find any of the hotels offer big family rooms. They are smallish and the bunk beds are no more than a foot away from the bed so you would have to all go to bed at the same time!!
Of course to reduce cost you could take the kids out of school and take a low season break. I would go to the Eurodisney site, put in the holiday of your choice and use that figure as a bench mark when approaching other companies. I would be surprised if you get it much cheaper.
Good Luck