My favourite one handed melee weapon has to be blood-nap, IMO the best looking knife which has a nice sneak bonus (which wierdly affects perk requirements) and has decent crit damage for such a fast weapon. Arguably the best knife aswell, a perked chances knife may do more damage but this has double the crit damage. The unique GRA shiahkebab is easily the strongest 1h melee weapon with the pyromaniac perk.
Favourite two hander would be the blade of the east, but the strongest would have to be old glory for a crit character, oh baby for a non-crit character.
Favourite unarmed is probably love and hate, fast punch speed so you get a lot of uppercuts and crosses in combat, looks badass and is easy to get. Strongest is debatable, I prefer the ballistic fist at the moment but there are plenty of contenders, the industrial hand, embrace of the mantis king, super-heated saturnite fist etc.