In pathfinder there is a hybrid class called the brawler. It is part monk, part fighter. It can increase the damage of "close weapons" to the damage of an unarmed strike from a brawler 4 levels lower. What weapon would allow a brawler the greatest damage output? The main points being a weapon whose damage will be increased and has a good critical range.
People far better with numbers then I have created the following formula to calculate average Damage per Attack. Check out this Thread for more information regarding the formula. Check out this AnyDice program for more information and calculations as well.
The damage formula is h(d+s)+tchd. h = Chance to hit, expressed as a percentage d = Damage per hit. Average damage is assumed. s = Precision damage per hit (or other damage that isn't multiplied on a crit). Average damage is again assumed. t = Chance to roll a critical threat, expressed as a percentage. c = Critical hit bonus damage. x2 = 1, x3 = 2, x4 = 3.
Assuming you have a melee attack bonus of +11 after all modifiers have been applied, against an enemy with AC20, we can fill this out for your unarmed strike as follows
Which equates to 1.925 damage per attack (Not including any strength bonus)
The Fighting Fan or Punching Dagger, both increase your critical multiplier to x3, which equate to an average DpA of 2.1175
The 19-20 ciritcal range of the Cestus gives the same result of 2.1175.
Your unarmed strike at 1d8 damage provides a result of 2.59875
The Scizore is a close weapon which deals 1d10 damage at the cost of 1 to your attack roll, and provides an average result of 2.8875 DpA
This result shows that while you are able to increase the damage of any weapon which does less then 1d6 damage, you are still better off using your unarmed strike, and better again with a weapon with a higher damage dice.
At level 5 with the brawler proficencies, I would suggest a Scziore as the best option for dealing the highest Damage per Attack possible.
Though a Close weapon with an 18-20 critical range, or an x4 critical multiplier does not exist, if one did for any reason, it would have an average DpA of 2.21375 as a comparison.
Close Weapon Mastery
At 5th level, a brawler's damage with close weapons increases. When wielding a close weapon, she uses the unarmed strike damage of a brawler 4 levels lower instead of the base damage for that weapon (for example, a 5th-level Medium brawler wielding a punching dagger deals 1d6 points of damage instead of the weapon's normal 1d4). If the weapon normally deals more damage than this, its damage is unchanged. This ability does not affect any other aspect of the weapon. The brawler can decide to use the weapon's base damage instead of her adjusted unarmed strike damage—this must be declared before the attack roll is made. (Source, emphasis mine)
The ability you're speaking of doesn't increase the damage of all close weapons. It instead brings those close weapons that deal damage lower than your unarmed strike (or, what it would be at your level - 4) up to that of your unarmed strike (Lvl - 4).
So the weapon that would grant your brawler a greater damage output would be any that deals damage greater than your unarmed strike at lvl - 4. Using the example in the rules text, this would be any weapon that with (a) damage di(c)e greater than 1d6. It's meant to make those weaker weapons more viable and not to increase the damage of weapons that already deal more than that.