The cookies are going stale because of a change in moisture of the starches in the cookies. It's something along the lines of processing the moisture in the cookies to crystallize the starches, which is what leaves the cookies with that dry, hard, staleness. The solution to fix this effect, is to provide another source of moisture, so that the starch can process that moisture instead of taking it into the cookies.
So what you need to do, is to provide the cookies with a source of moisture. If you place a couple slices of bread into the container with the cookies, and by some sort of magic the moisture from the bread will transfer to the cookies.
I've even used this to soften cookies I accidentally over baked, making them nice and soft after a day of re-moisturizing.
I don't know how long you can keep cookies fresh with this method, but I've definitely kept them for over a week by replacing the bread slices as they became stale.
I have heard that apple slices will do the same thing, but I've never tried it because I don't like the idea of apples rotting in the same container as my cookies.