I'd like weapon damage to be a bit more linked to character Hit Points, in a simulationist sense.
For example, in Pendragon RPG, a character's damage is equal to his/her Strength+Size/6 in d6s. So if a character has a Strength of 14 and a Size of 12 (14+12/6) it equals 4d6 (round down). If you then understand that the HP total is equal to Constitution+Size, then you can already see a relationship in a simulationist sense.
Now D&D Basic actually did this in a way. The Hit Dice used to generate HP per level where actually equitable with the damage levels of weapons. A Fighter, for example had a HD of D8 per level, and usually dealt out damage of D8 with a sword. A Thief would have a D6 HD per level, and usually carry a Short Sword (damage D6), A Wizard had D4 HD and carried a dagger (D4), and so on.
It wasn't exact, but actually, I think there is some notion that if Class HD goes up per level, they are essentially able to take a number of eqitable combat blows equal to their respective level before they are likely to drop.
Or something like that anyhoo...
Beyond that, you could simply classify damage types into polyhedral dice types:
D4 Unarmed Brawling
D6 Daggers, Sling Shots, Darts
D8 Short Swords, Maces, Morning Stars, Spears, Arrows, 1 Handed Axe (throwing), Hammers
D10 Swords (Broad or 1 handed Long), Long Spears and other polearms, Heavy Axe.
D12 Two Handed (Bastard) Swords and Two Handed Battle Axes.
Howzat?
The Hit Dice per level could tally in with these, perhaps.