Adding a new dimension to D&D weapons

D&D weapons have many different traits, depending on the edition. Usually, damage is the most noticeable. If I give my players a list of weapons, they'll usually pick the most damaging weapon they can carry.

But there is also price, weight, and in some editions speed, effectives versus armor, length, size, and different effects due to weapon proficiencies.

There is ONE dimensional that is sorely lacking since OD&D, however: an attack bonus.

In chainmail, weapons cause the same damage, but the to-hit chance is different depending on the weapon and armor. This is NOT simply an effect of armor: some weapons are just better to-hit overall.

For example, giving B/X battleaxes a +2 to hit would be very easy. They'd become better than swords against plate and shield, but still worse in most circumstances. Maybe 2h-swords deserve a +1 to-hit to compensate the "slow" aspect (if you're using that at all). Maybe maces deserve a +1 bonus (and make them weight a bit more than short swords). Etc.

What about unarmed attacks? They could cause 1, 1d2 or 1d3 damage... But it would be just as easy to give them a -2 penalty to-hit. Now attacking a foe in plate with bare hands is very ineffective.

[Notice that, because of how AC works, the bonus would be greater for armor-defeating weapons; these are usually heavy, unbalanced... In short, this is the opposite of speed.]

You could scatter high-quality and low-quality weapons by giving them -1 and +1 to-hit... without altering damage output significantly (probably less than 20%, or a smaller difference than 1d6 to 1d8 damage).

There is almost no effect on game balance.

This is probably the simplest addition you can make to B/X weapons. 

You don't need special rules or subsystems, and you probably can write your attack bonus (if you're using that) with your weapon - which already includes Strength/Dexterity and class bonuses. E.g.: "Dagger +3 [1d4+1], Polearm +6 [1d10+1]".

Unfortunately, the "+1 magic weapon" idea is too strong in D&D; it is expected that weapons only give you bonuses to hit if they are magical (it would be a good idea to mix things up here too: a magical weapon with +2 to-hit and +4 damage, for example, something 5e has embraced).

But giving ordinary weapons discreet bonus/penalties to-hit would already add an entire new dimension to your arsenal.

This is just a random idea for now; but if I ever had to add weapon details to my "minimalist B/X", this is where I would start.