Okay, I have been working on this in my spare time, cobbling all the bits and pieces together as I go.

Weapon Damage
Weapons have their base damage (listed below), but standard melee attacks actually roll two dice for damage (so a longsword does 2d8+1 damage, a dagger does 2d6 damage, and so on).  This is done for two reasons.  First, it allows me to split abilities into multiple attacks (which will come later); second, it lets everyone roll multiple dice for damage (which I like); and third, it keeps base weapon damage more relevant as the game goes on.

Weapon Sizing
A weapon's size determines who can use it and how he uses it.  

• A creature can wield a melee weapon of his size category in one hand or two hands.  When wielded in two hands, such a weapon gains a +1 bonus on damage rolls.
• A creature can wield a melee weapon one size category smaller than he is in one hand.  He cannot effectively wield the weapon in two hands.
• A creature can wield a melee weapon one size category larger than he is in two hands.  It is simply too large to wield with a single hand.

Ranged weapons are a little trickier.  Crossbows and slings follow the same rules as melee weapons because they need a single hand to operate (although they can be steadied with two hands to give them the same +1 bonus on attack rolls).  

Darts are always wielded in one hand, and they can be used by creatures of their size category or up to one size category larger (so Small and Medium creatures can use them normally, though they're not that great).

Bows are always wielded in two hands, and they are more difficult to wield.  You can wield a bow up to your size category or one size category smaller, but you cannot wield a bow that is a size category larger than you.  You receive no bonuses on your damage roll from wielding a bow in two hands.

Dual-wielding: Adventurers who wield two melee weapons in combat get a +1 bonus on melee attack rolls that they make.  They may choose to attack with either weapon.  Neither shields nor unarmed strikes can be used to dual-wield.

Weapon Properties
Double: Double weapons are weapons that grant all the benefits of dual-wielding when they are wielded in two hands.

Finesse: When using a finesse weapon, the wielder may add his Dexterity bonus to attack and damage rolls in place of his Strength modifier.

Heavy: You add your Constitution modifier to all damage rolls with heavy weapons (in addition to your Strength modifier).

Reload: Ranged weapons with this property must be loaded with ammunition after every attack made with them.  The effort required to load them is given in parenthesis.  For instance, reload (free) means reloading the weapon is a free action, while reload (move) means reloading the weapon is a move action.

Reach: Reach weapons increase their wielder's natural reach by five feet, allowing them to strike targets further away.  Small and Medium creatures have natural reaches of five feet, so those wielding reach weapons can hit opponents up to ten feet away with a reach weapon.  (Large and larger creatures typically have reaches of ten feet or more, so reach weapons wielded by them increase their reach to fifteen feet or beyond.)

Thrown: You can easily use a thrown weapon as a ranged weapon.  When using a thrown weapon to make a ranged attack, you may make a melee attack with a range increment of 20 feet.  Weapons without the thrown property are treated as improvised weapons when thrown.

Impaling: When the wielder of an impaling weapon scores a critical hit, he cripples his enemy.  Until the end of its next turn, the creature takes a -2 penalty on all attack rolls and saving throws (including any made against effects or poisons delivered by the critical hit) and is flat-footed.

Cleaving: When the wielder of a cleaving weapon scores a critical hit, he deals a vicious wound to his opponent and then carries through the attack on another.  The creature initially struck by the critical hit has his speed reduced to five feet until the end of its next turn.  Another creature within reach of the wielder automatically takes 1d8 + half the wielder's level + the wielder's Strength modifier damage.  This damage increases to 2d6 + half level + Strength modifier damage at fourth level.

Crushing: When the wielder of a crushing weapon scores a critical hit, he briefly stuns his enemy.  The creature struck by the critical hit is dazed by the blow and loses its next move action as it recovers.  In addition, it is forced backward a number of feet equal to half the damage dealt by the attack.

Now, as far as critical hits go: everything crits on a 20.  Crit ranges otherwise do not exist because they're a pain in the ass to track.  Now as far as weapons go:

Simple Melee Weapons
• Dagger (small) - 1d6 dmg - finesse, impaling, thrown.
• Short sword (small) - 1d6+1 dmg - finesse, impaling.
• Club (medium) - 1d6+1 dmg - crushing.
• Mace (medium) - 1d8 dmg - crushing.
• Sickle (small) - 1d6 dmg - cleaving.
• Spear (medium) - 1d6 dmg - reach, impaling, thrown.
• Quarterstaff (medium) - 1d6 dmg - crushing, double.

Simple Ranged Weapons
• Dart (small) - 1d4+1 dmg - impaling.
• Crossbow (medium) - 1d8 dmg - impaling, reload (move).
• Shortbow (small) - 1d6 dmg - impaling, reload (free).
• Sling (small) - 1d6 dmg - crushing, reload (free).

Martial Melee Weapons
• Handaxe (medium) - 1d6 dmg - heavy, thrown, cleaving.
• Battleaxe (large) - 1d8 dmg - heavy, cleaving.
• Morningstar (medium) - 1d10 dmg - crushing.
• Flail (medium) - 1d6+1 dmg - heavy, crushing.
• Glaive (large) - 1d8 dmg - reach, cleaving.
• Greatsword (large) - 1d10 dmg - cleaving.
• Longspear (medium) - 1d6+1 dmg - reach, impaling, thrown.
• Longsword (medium) - 1d8 dmg - cleaving.
• Ranseur (large) - 1d8 dmg - reach, impaling.
• Rapier (medium) - 1d8 dmg - finesse, impaling.
• Pick (medium) - 1d10 dmg - impaling.
• Scythe (large) - 1d10 dmg - cleaving.
• Scimitar (medium) - 1d8 dmg - finesse, cleaving.
• Warhammer (medium) - 1d6 dmg - heavy, crushing.

Martial Ranged Weapons
• Longbow (medium) - 1d8 dmg - impaling, reload (free).

If you can think of any more to add here, I'd be interested in hearing about them.

Miscellaneous Weapons
• Unarmed strike (small) - 1d4 dmg - none.
• Improvised weapon (small) - 1d6 dmg - none.
• Shield bash (small) - 1d6 dmg - crushing.

Note on miscellaneous weapons: you are not considered proficient with them (thus taking a penalty on your attack roll) and their size varies.  The listed damage above is based on the damage done by a Small unarmed combatant, a Small shield basher, and a Small-size improvised weapon.  Unarmed strikes and shield bashes naturally scale upward depending on the size of the attacker (Medium unarmed strikes/bashes do 1d6 damage, Large unarmed strikes/bashes do 1d8 damage, and so on).  Improvised weapons depend entirely on the relative size of the weapon and scale from that point onward.

I know this has never really been captured in D&D, but one of the key things that differentiates weapons like swords from weapons like axes is that sword are generally designed to defend as well as attack while axes are not, and spears and staffs (because of their wooden hafts) can be used to defend but will likely be destroyed blocking a heavily-bladed metal weapon.

If you wanted to capture that, you could give a +1 to AC for weapons that assist in defense while giving those that don't a bit more damage such that a character can choose to boost their AC a little or do more damage.  I can think of a few ways to handle spears and staves.  The simplest would be to give the AC bonus only while fighting against weapons that can't destroy the haft if it's used to block or to allow the AC bonus for weapons that could break the blocking weapon but only break the weapon on, say, the first successful hit.