In the original D&D (the boxed set from 1974), all weapons caused 1d6 damage. Variable weapon damage (with d4s, d8s, etc) was introduced by Greyhawk and has been widely adopted since then, not only by all editions of D&D since, but also by different systems such as Call of Cthulhu and Savage Worlds. Nowadays, most RPGs differentiate weapons by the damage they cause, even when there are few mechanical differences otherwise.
Using the same damage for all weapons might work fine, but there are some caveats. If only cost and weight vary by weapon, adventures would always use the cheapest, lightest weapon they can find, which is a bit odd. But the same will happen in other systems where weapons have a single variable that is more important than others. For example, when cost becomes unimportant in B/X, everybody will use the weapon with the greater damage, unless weapons you use other traits for weapons, such as reach, speed, handedness, proficiency, effectiveness against armor and shield, etc.
To account for this, AD&D has a complex weapon system involving lots of special rules, exceptions and tables – a system which was completely abandoned not only in subsequent editions but also ignored by many AD&D players at the time and even disowned by Gary Gygax himself (“Actually, yes, as I wanted to remove some things from the AD&D rules--weapon speed factors, weapon vs. armor” - http://greyhawkgrognard.blogspot.com.br/2009/02/ad-second-edition.html)
To be honest, I love the idea of weapons with different weight, damage, speed, reach, etc. While I dislike the fact that this often comes at the cost of simplicity, dread using tables to find the numbers I need, and I can see the argument for D&D combat being too abstract to worry about such things, I enjoy this little differences anyway. And detail should be where your heart is, after all.
In my own game (Days of the Damned), I tried to keep weapons diverse but simple. They might have many different traits (damage bonus, speed, weight and cost, etc.), but they are often the same number, so you don’t need to check attack tables or long lists of traits for your weapon.
The system below is an adaptation to OD&D and BD&D, using only the d6, as the original game intended. The same system can be adapted to different systems that use the a similar rationale for weapons, such as Savage Worlds.
First, divide the weapons by Size:
Tiny (-2) weapons are small, concealable weapons such as daggers, saps, etc.
Medium (0) weapons are the default “medium”, one-handed weapons.
Great (+2) weapons are two-handed weapons such as dopplehanders, great-axes, and so on.
Obviously, medium weapons deal 1d6 damage, tiny weapons 1d6-2 (minimum 1), and great weapons 1d6+2. But that’s not all they do - this numbers are also used for all other purposes, depending if the size and weight of the weapon will help or hinder you.
Here are some examples:
Speed: Bigger weapons get +2 initiative bonus (because of reach) until the wielder is hit for the first time; after that, smaller weapons get the bonus (because they are lighter and faster). In confined spaces, smaller weapons always get a +2 bonus, and great weapons a -2 penalty. This assumes initiative means something in your game; if this isn’t the case, apply the bonus to hit, instead, or use some formula to control reach (attacks from second rank, opportunity attacks, etc.).
Backstab and grapple: Likewise, while grappling or attacking by surprise, all weapons deal 1d6 damage, but smaller weapons get a bonus to hit, while bigger weapons get a penalty. The same happens if you’re trying to conceal or smuggle a weapon.
Big creatures: Great weapons get a damage bonus against big creatures, tiny weapons get a penalty. I would limit this to +1/-1; you still wouldn’t want to hit a dragon with your knife.
Critical hits: If you use any rule for critical hits, they should add 1d6 instead of double damage, so that even a knife can be deadly sometimes, especially with a well-placed blow.
Armor: this is a little trickier. If you want “weapon vs. armor” to matter, there are multiple ways you can go about it. For example, you can divide armors in three groups: light/unarmored, medium/shield and heavy. Depending on the weapon, it gets a +1 bonus to hit light/unarmored and -1 penalty to hit heavy armor, or vice versa. Some weapons, like maces, may get a +1 penalty to hit medium/shield, instead, while other have no bonuses or penalties at all or have only bonuses (see below).
Price and weight: In general, bigger weapons are heavier and more expensive (Days of the Damned uses 2 kilos for medium weapons, 4 kilos for great weapons, and half a kilo for tiny weapons, price $10 per kilo), but a little bit better overall.
Other distinctions, good or bad, can be added on top of that, or you can remove the ones you don’t like – just try to keep some balance between heavier and lighter weapons. The important thing is that not all players need to use these rules, even if some of them do. If a player is looking for something simple, 1d6 damage and no bonuses or penalties will do.
Besides, you don’t have to use all traits in your game – just choose some traits that make big weapons better in some circumstances and worse in others. I would still give a small edge to big weapons because they are heavier and more expensive.
I usually let players add two traits to any weapon: reach, thrown, double damage on a charge, parry, grapple, shield-breaker, improvised, bonus against armor, etc. Extra damage is +1 per trait (maximum 1d6+3), other traits are usually +2 (maximum +4). A pike (great weapon) could have damage 1d6+2, Reach +4 (and thus -4 in tight spaces) and charge, for example. If you do the same, players who want simple weapons can pick a 1d6+2 medium sword (with no other bonuses or penalties), a 1d6 medium spear that can be thrown and attack form the second rank, or a 1d6 medium mace that ignores shields and can be thrown. This is what it looks like:
Spear (0): reach, charge.
Sword (0): 1d6+1 damage, parry.
Great maul (+2): 1d6+3 damage, +1 against medium/shield or heavier.
You can add a few negative traits too (expensive, heavy, etc.) to create more combinations, although one might be careful to avoid abuse (and headaches).
You notice there are no 1d6+1 or 1d6-1 weapons. You can use this numbers for bucklers, main gauches and other weapons that can be used with the offhand (1d6-1) and as bastard swords and heavy spears that can be used with one or two hands (size +1, but causes only 1d6 damage if used with one hands).
In any case, you should avoid adding multiple modifiers to the same things. Use one modifier for initiative, one modifier for damage, one for attack rolls, etc. I also ignore differences in reach unless they are greater than +1. So, no reach bonus for the great sword against the bastard sword. This makes things a lot simpler.
But what about the other dice? Here is where it gets interesting: use them for magical weapons. An 1d8 sword will feel more “magical” if you physically pick a different dice than everybody else is using, even though the effect is mathematically similar to 1d6+1 (you still might add a bonus to BAB when using a magical sword, but you don't have to). What about an 1d8 sword, 1d10 versus undead? This feels really cool, especially if the improved damage extends to crits (which I do). You can use your d7s if you want something a bit more exotic, of course. And you can keep the "common" modifiers on top of it (a magical dagger deals 1d8-2 damage, for example).