Those mesh bags you can use for shopping commonly call for cotton. I'm not sure I'd use it for a real purse without a lining because of the stretch factor. These are great for using the same kind of rougher cotton you might use for a dishcloth.
Toys, coffee cozies, Christmas ornaments, baby booties, baby bib, coasters, doll clothes, placemats, bowls, baskets - all are good cotton options.
There are tons of good hat patterns designed for cotton as well. I'd want to use a mercerized cotton for those for softness, though.