The key to a "cultural weapon" is that the weapon and the culture evolve in parallel, interacting with one another. To make one realistically, you will need to evolve them in parallel as well.
This means that, as the weapon evolves, it is allowed to shape the culture in a way which then, in turn, shapes the evolution of the weapon.
One trick you can do to help identify cultural weapons is to do perturbation analyses. Tweak your culture just a little (not a lot), and see what changes. Do this over and over, and you'll find that some facets do not change very much. You'll find cultural weapons appear along these lines, because the culture will always be having to bend and flex (as all cultures do), but a weapon earning the title of "cultural weapon" will be along an axis that doesn't flex.
For example, take the katana, weapon of the Samurai. One of the unchanging facets you will find when you perturb Japanese culture is their unrelenting efforts for perfection of their art. Accordingly, the katana is a weapon which is designed for a warrior who has sought perfection. It magnifies the perfections in the Samauai code of bushido, but is very unforgiving to imperfections.
You can also look at the mongol horde. They are famous for basically inventing horseback archery. This one is easy to see: no matter how you perturb the mongol culture, they always find their way to better locations on hoseback.