Weapon evolution

Do you mean weapon refines?
For inheritable weapons and some prfs you can go to the weapon forge (once you've unlocked it in the story) and exchange refining stones/arena medals (both from the arena) to gain a stat boost or an effect from them. Some weapons (namely the Wolf tomes and the Killer line of weapons) take a small amount of those two materials to upgrade to a better version (Keen Wolftomes and the Slaying line) that have a bit better Might. And from there then you can refine them for a stat boost or for the effect.

If you don't mean weapon refines then I have no clue, sorry.

A weapon evolution is when a weapon is similar or is the same exact weapon appearance to it's original but how the weapon works is different. For example Siegmund has a hone attack 4 and a brash assault for it's refinement or you can evolve it into flame Siegmund and make the weapon have attack +3 and a follow up attack if you're far away from allies.