Damage Dice!
Idea for the current path my game, Tales of Mythic Prehistory, is taking towards a standalone second edition.
Each character has a damage die. Some characters might have multiple dice and use that in fun ways, as shown later in this post. More martial characters have bigger damage dice, but almost no one has actual static bonuses to their damage.
I imagine Weapons either applying effects if they deal 6+ damage in a single hit or providing some other ability to their wielders. A club stuns the enemy on a 6+, while a sword lets you parry one attack per round to reroll incoming damage.
An example ability using multiple dice. I imagine this character starting with 2d4 Damage.
Stunts: On a successful attack, roll any number of your damage dice and inflict that damage to the defender. Perform one Stunt for each damage die you didn't roll.
Boast: Gain a D4 Mighty Deed.
Taunt: Stun the defender.
Tumble: Move NEAR.
I used a version of this model in my First Contact game, and I think it's worth exploring more.