Well, make sure its nothing you want to reuse
With Brie, one of my girlies...
I gave her a wheel (simple clip onto the cage), a teabag box, a kitchenroll tube, a pile of tissue, loads of food and so on...
You wouldnt believe the nest that hamster constructed! She removed the wheel from the cage, turned it up and filled it with some nesting material, then put the box in the wheel, filled that with nesting material and then manage to get the tube into the box and wedge it against the bars at the top of the cage so the nest didnt fall over.
She then proceeded to put every single bit of bedding and shavings into the box and wheel and then wedge herself into it somewhere.
Amazing what you can do with household items!