This is an idea for weapon training that I had which is probably too fiddly for online play, but might work in person. To gain more than a +1 to attack or damage (whether from attack bonus, strength, or wherever bonuses come from in your system of choice), a combatant must train with a given weapon. Each weapon is rated with minimum stat requirements and maximum bonus potential. Mundane weapons might be, for example, min +0 and max +3/+3, meaning that anyone without a penalty can use them and they can support at most a +3 to attack and damage. Insisting on using a weapon without the minimum stat imposes some large penalty (-4 or 5E style disadvantage).
The constraints would need to be tracked for each weapon along with training level, which could default to starting scores for starting weapons initially (representing background training). For example, consider a first level Labyrinth Lord fighter with 16 strength, which grants a +2 to attack and damage. A first level fighter in this system also has the equivalent of a +1 bonus to attack. Assume further that this character starts with a battle axe and dagger. Under skills, the player would write battle axe +3/+2 and dagger +3/+2. This is the character’s initial training. If a sword is picked up, it will be wielded with +0/+0 until the character can train with it.
How does training work? As a downtime action, the character can pay for training in a weapon. This costs some set amount, maybe based on bonus to be unlocked, say 500 GP per target plus (so, moving from +1 to +2 would cost 1000 GP). This raises either the attack or the damage by one point, assuming that is supported by class attack bonus or ability scores. Thus, that example first level character above can gain no more bonus points in battle axe until ability scores or attack bonus increase.
The benefits are that it gives fighters something to do during downtime actions, somewhat restrains bonus progression, makes special weapons more valuable without needing to resort to magic weapons as treasure all the time, and allows fighters to ease in to using crazy weapons like the Berserk dragon slayer sword or the asylum demon’s great hammer which might make their use feel a bit more special (and also more fictionally justified). It would also allow weapons to be numerically defined on dimensions of finesse and brutality. For example, a big club might be +1/+4 in potential, making it a good choice for a character with low skill but high strength (assuming such is possible in the base system).