What do I gain for completing weapon side-missions?
Solution 1:
By completing weapon side-missions, you gain Mirian (currency to upgrade stuff) and when you complete enough missions, the look of your weapons will change.
Some people mentioned, that you also get a few more execution animations and you break shields more easily, but that is only conjecture.
Sources:
Gosunoob.com - Sword legend
GameFAQs thread