Weapon damage is kind of a contentious thing in D&D. There are quite a few different methods, and probably dozens if not hundreds of tweaks and variations on each.
There's straight 1d6 for everything (possibly "roll twice, take higher" for big two-handed weapons.) This has simplicity on its side, but it also makes choice of weapons pretty much pure fluff.
There's the optional variable weapon damage rule from B/X, which feels better intuitively to me, but also makes it incredibly obvious which weapon is "best." (It's the one with the highest damage die, possibly mitigated by the inability to use a shield.) Likely to result in Every Fighter Wields a Sword.
There's damage by class or Hit Die. Fighters and dwarves always do 1d8. Clerics, elves, and halflings do 1d6. Thieves do 1d6 or 1d4, depending on the DM's concept of the class. Magic-users do 1d4. It does a good job of enforcing the combat supremacy of the fighter class, but once again, the actual choice of weapon is merely one of style.
There's the Weapon Mastery system of BECMI. It looks cool on paper, but in practice it can be a bit of a nightmare, the huge damage increases and other benefits can be wildly unbalancing, and a character who specializes in a single weapon will tend to be vastly superior to one who studies more broadly.
There's the weapon proficiency and specialization of AD&D, the primary effect of which seems to be giving fighters a damage bonus with a chosen weapon.
Some of those are closer to my ideal than others, but none is completely satisfying. What I want is something that's simple, makes weapon choice matter, and makes fighters better at dealing damage. Here's what I've come up with:
1d2 damage unarmed
1d4 damage for light or crude weapons, such as daggers, clubs, hand axes, and slings
1d6 for medium weapons, including all other one-handed weapons plus the quarterstaff. Note that this includes the traditional "normal sword," which is no longer the obviously superior choice for every fighting man, as well as the battle axe.
1d8 for heavy weapons that must be wielded two-handed, such as polearms and two-handed swords
Fighters use the next higher die for all weapons. A fighter thus does 1d6 with a dagger, 1d8 with a sword or mace, and 1d10 with a two-handed sword.
Dwarves (if you're using B/X race-classes) step up one die for traditional dwarven weapons. In a traditional fantasy world, that's probably axes and hammers.
Elves step up one die for traditional elven weapons. Sword, spear, and bow are typical.
Halflings step up one die for traditional halfling weapons. As far as I'm concerned, that's only the sling. Hey, they may be decent fighters, but halflings aren't supposed to be damage-dealing juggernauts.
All other classes use "allowed" weapons at base damage. If using a weapon not allowed for their class, they drop one die size. (Clerics may suffer other penalties for using proscribed arms, but that's another matter.) For example, a magic-user using a sword (a one-handed 1d6 weapon) would do 1d4 damage. With a two-handed sword, he'd do 1d6. He can do so as a matter of style, if the player likes, but it's not going to gain him any advantage over using a dagger or staff.
Weapons that are normally used one-handed but can easily be used two-handed, such as clubs, battle axes, and bastard swords, step up one die size when wielded with both hands. This is cumulative with fighter bonuses. A fighter using a club two-handed would step up two dice, to 1d8.
Fighting with a weapon in each hand grants +1 to damage. The second weapon must be a light one.
The overall effect comes pretty close to the class-based damage model, but allows some room for weapon choice to have an effect.
To distinguish weapons a little bit more and make weapon choice more meaningful:
Swords do +2 damage to unarmored and lightly armored opponent, i.e. leather or less. Monsters with AC6 or better that isn't due to agility and dodging are considered heavily armored for this purpose.
Heavy impact weapons such as maces, war hammers, battle axes, and various polearms gain +2 to hit vs. chain mail or heavier armor, representing their real-world use in defeating armor.
Spears do +2 damage to large opponents, due to their ability to penetrate deep into flesh, where other weapons have difficulty doing more than superficial damage.
Quarterstaves grant a bonus to AC as if the wielder had a shield.
Crossbows are +2 to hit everything, but fire every other round.
That's it. Not overly complicated, I hope; at least it doesn't offend my B/X sensibilities too badly. Next time I get a chance to run a game, I'm going to give this a whirl.