Suppose the weapon is a flute. Naruto could use it to control summons or even manipulate the chakra flow in his surroundings. He might be able to play certain tunes that can either heal his allies or put enemies in a trance. This would add a whole new dimension to his abilities and make him a more versatile fighter.
Let's say the weapon is a shield. This would change his fighting style completely. Instead of always being on the offensive, he can now also focus on defense. He could use the shield to block powerful jutsus and then counter - attack when the enemy is vulnerable. It will also give him more confidence in battles against stronger opponents.
If the weapon is a blade, it will probably make his taijutsu more lethal. He can combine his already impressive speed and strength with the weapon's sharpness. His attacks will be more precise and deal more damage.