You'll want to avoid anywhere damp, since moisture is an enemy of cardboard. So a basement that isn't dry, would be a bad choice. Mice won't seek out your cardboard, unless there's food in it. They might chew it and use it for nesting material if they happen across it, but won't make seeking out cardboard a main part of their day.
I've had cardboard in the loft of my garage for over 5 years, with no problems. I live in NY, so hot summers (80-90F) and cold winters (0-20F). The garage is detached and unconditioned, so it experiences hot humid days in the summer and bitter cold in the winter.
Cardboard is fairly hardy, and will last quite a long time if kept dry. So anywhere that is out of your way and dry, would be a fine place to store it.