It would seem at first that the mere mass of a weapon would be a great reason to vary damage by weapon type. After all, if bigger weapons don't do more damage, why would anyone buy them? They weigh more than light weapons and are thus more encumbering. But thing again about the foundations of D&D combat: an ordinary man-at-arms has 1d6 hit points, and a short sword does 1d6 damage. An ordinary short sword has a chance to kill an ordinary man. Should a heftier sword kill an ordinary man even more? Or should it have a better chance of scoring that potentially deadly hit?
This is the rationale behind giving more massive weapons a bonus to hit instead of a bonus to damage. We can treat mass as an intrinsic quality that provides a bonus where appropriate: figure that basic one-handed weapons like the short sword as the default weapon mass, give a +1 advantage to two-handed weapons and another +1 if the weapon is too heavy for a person of normal strength to wield effectively, but the fighter has high Strength. Also give the advantage to the side with the more massive weapon or shield, if parrying or blocking.
You can, of course, allow users of two-handed weapons to roll two dice and pick the highest. That's a common LBB houserule. I'm reluctant to do so only because I plan on letting everyone roll two dice and picking one for speed and the other for damage, but perhaps I'll work around that.