A couple weeks ago, I declared that I didn't like the damage algorithms that ask you to pick the smallest or greatest of two dice for small and large weapons respectively. I still agree with this, at least as presented, but I've come around to a variation.
Rather than distinguishing between small/medium/large weapons, just distinguish between one-handed or two-handed weapons. One-handed is normal, but if you wield a weapon in both hands (whether that means the weapon is two-handed or you're wielding one weapon in each hand) you take the best of two dice. That way, we are relating damage to how many hands are busy, so to speak. One hand, one die. Two hands, (best of) two dice.
Then, we can vary the die size itself by different factors, such as the wielder's size (rather than that of the weapon) or by the weapon's material, depending on what you're going for. This part is less important to me than the above, but I do think it's useful to have basically two axes of variation.
We can even use damage types to offer a third variable factor, impacting neither die size nor how many are rolled, but whether the opponent is resistant or vulnerable to the type of damage dealt. Maybe you're wielding a two-handed d8 warhammer and you roll an 7, but against a slime that's only as good as 3 hp damage.
Just brainstorming! Keeping track of handedness and damage type feels easier than having weapon tables or a bunch of separate qualities.