I can't remember who wrote it, but there was a blogger post years ago about how every pre-modern weapon developed, from the dart to the bohemian ear-spoon, was developed for a reason. There weren't "good weapons" and "bad weapons"--they had niches they were meant to fill and they did. It's nice because it makes players think about what weapon they're using, and so think about the fight in more detail.
I used this idea a few times--in Demon City, for instance, because it's a game based on horror movies, nearly all weapons, whether a .45 or a flowerpot to the dome, do the same initial damage, but weapons that are better in a given situation vs whatever the other guy has get an advantage to hit.
Somebody recently commissioned me to write a "light and old-school but more tactical" D&D-variant and I used this same idea. The assumption in this game is all weapons do around d6 damage based on the user's class instead of damage by weapon type (in original Warhammer, damage is based on strength) but each has one or more simplified gamey rock-paper-scissors-style feature that make them better in certain situations. So instead of a flail making it easier to get past shields, a flail always ignores shields.
Some notes on these notes:
This game also includes a "block" option, which includes parrying or the like, so in games with no separate parrying step the rule would have to be slightly different.
If a weapon offers a bonus to block I'd usually just translate that into a +1 ac if the user is trained (aka a fighter).
Where it says "bonus" read "+2" in D&Dlike systems.
Cestus: Bonus to hit and damage at hand range (up to arm's length). Minus to both any other time.
Chain: Ignores blocks and shields. Maximum 4 damage. Can be used to grab or grapple.
Club or Mace: Bonus to knock an enemy in heavy armor prone.
Dagger: Throwable. +2 to hit and damage if used in a melee sneak attack. Maximum 4 damage otherwise.
Dart: Bonus to strike at throwing range (30' ish).
Flail: Ignores shields.
Garotte: Can only be used as a surprise or on a grappled enemy. Ignores armor. (Enemy must have a neck.)
Great Axe: +2 damage. -1 to hit.
Greatsword Can’t be blocked by spears, staffs, pole-arms, can't be used in close-quarters. In a game without a block or parry I'd just make this work like the great axe.
Hand Axe: Throwable.
Heavy pole-arms (halberd, pike, etc): Bonus to damage at spear range (10-12') An extra block per round at spear range. Can’t be used in close quarters.
Longsword, Shortsword, Scimitar: Bonus to damage and block at sword range.
Net: No damage. A net can be thrown up to 30’ and acts much like a grab on the target’s legs and can be escaped using the same kind of rolls. If the attacker doesn’t hold onto the net (like, they’re killed that round, for example) it can be escaped in one round.
Rapier: Bonus to strike unarmored or leather-equivalent-armor opponents.
Sling: It's basically free and doesn't look like a weapon. Maximum 4 damage but can hit two adjacent opponents at once if you sling a handful of rocks.
Spears and tridents: Throwable. An extra block per round at spear range. Can’t be used in close quarters.
Staff: Doesn’t look like a weapon. An extra block per round at spear range. Maximum 4 damage. Can’t be used in close quarters.
Lance: Bonus to knock a mounted enemy prone. Minus to hit if not mounted.
Warhammer: Bonus to knock a mounted enemy prone.
Whip: Harmless against armor heavier than leather. Maximum 3 damage. Can be used to grab or grapple at a bonus.
For bows: I stuck with the basic D&D idea. The light crossbow is the standard, the shortbow gives you two attacks per round but less damage, the heavy crossbow gives you the most damage but shoots every other round, the longbow gives you more damage than the light crossbow and the best range, but can't be used in close quarters.
To keep it light, I didn't go nuts with the different kinds of swords (sabres and scimitars supposedly being better from horseback, khopesh swords being better for hooking limbs, etc).
For a non-western game I'd add the shuriken does less damage than a dart but can be thrown twice per round and the indian chakram does less damage but has a better to-hit.