I often throw empty Salada boxes and Rice bubble boxes on top of Burts cage for him to rip up. You will need to double check the boxes yourself and work out what is good/safe and what is not- I do not use any freezer boxes as they do have plastic coating on them, every now and then my birds get given a full loo roll to shred (I do not buy scented nor printed paper), some boxes from parcels I will also give them to chew. Pen lids and marbles used to be a favorite of some of my past budgies, Burt loves clean empty drink bottles and plastic cups, SS utensils like serving spoons and even normal size spoons. You will need to judge each and every item you want to use and supervise to make sure your bird interacts with it the way you want them to but you will find many things that are suitable.
Not sure what kind of bird you are asking for but there are many things found at home suitable for many pet birds.