I would do a little research and try to find out what someone would have used in that time period. If you can't find any information like that then you could do wool or another fabric that picks up one of the colors on the front. If I remember correctly, though, crazy quilts were brought into vogue originally by ladies that were able to afford fancy fabrics. Maybe even a nice piece of velvet would be a nice touch for the backing.
I would probably want to choose a lower loft batting like warm and natural. I think that would be more period appropriate but would still look a little bit contemporary. Wool batting is going to give a nice puffiness and define the stitches nicely but depending on the blocks the quilting might be lost. As for the binding, I would do a knife edge binding; nice and clean.
I'm not sure about the quilting. Do you have a pic that you can post?