One thing I've always had trouble with when building a world is developing weapons that are unique to the culture in which they appear. The longsword is a staple of European armories, while India developed the bagh naka and the Ottomans had the yatagan. Unique weapons give a culture additional depth and an exotic flavor.
What process can we use to create cultural weapons?
Weapon shape and design is a mixture of many properties. A combination of technology, available material and culture. Does your world have a material that lends itself to hold its shape if it is long such as steel (long sword), cannot be long such as bronze (short gladius), or tiny such as obsidian shards (Incan club).
Technology shapes the weapon. The long sword comes from the need to apply more force to a strike when going against armored opponents. The relative light katana is a product of advanced metallurgy which itself comes from lack of metal in the area.
The scimitar is a cultural construct from the steppe clans where every grown man had a horse.
Finally, culture will influence every part of the weapon's decoration. Pommels, handles, engravings and other shapes will have items particular to that culture.
Weapons design is to a large extent, determined by what you have to attack. If your enemies are on horseback, you will need a weapon with long range or extended reach like a polearm or a 6' long two handed sword to effectively engage someone on horseback. Katanas were developed not only because the process of getting high quality ore was difficult, but also because most opponents would be protected by laminar armour, so a light slashing weapon effective against a fast moving opponent was required.
Weapons also evolve based on the evolution of defensive technology. Longswords actually were of limited utility by the 1300's as heavier chain and plate armour became more common, leading to the development of crushing or smashing weapons like maces and war hammers, and swords evolving towards narrow, diamond cross sectioned weapons suitable for stabbing into the gaps of armour, so what was "cultural" in 1300 will have been discarded and long out of service by 1500.
Once you have the social, economic and technological backgrounds laid out (determining what sorts of enemies that might be faced, how many soldiers or fighters could be supported and how well they could be equipped), then the typical weapons can be determined for the setting.
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.
People living in jungles will benefit from short range weapons and substances derived from the vegetation. (blowpipes and poison).
People living on the plains will need a long distance weapon (bows and arrows or boomerangs) to reach fast running prey.
The Judogi-grasping moves of Judo came from Japan because of the clothes that people wore every day.
If you fill out the details of clothing, culture, wildlife and environment then, with luck, the weapons will follow naturally.
