I had an idea while I was driving today, and I decided that I needed to write it down as soon as I got home.
It's based generally on the Old School notion of rolling 3d6 in order for attributes, but pushes things even further.
We all know that it's typically better to roll an 18 for strength than a 3. A high strength means more damage dealt in combat, it means better modifiers when trying to do things where having innate physical force is advantageous. The same generally applies to most other stats too. A high roll means a greater capacity to influence the outcome of events when the appropriate attribute is brought to bear on the situation.
Over the course of six rolls for the attribute array, the law of averages said you're likely to get a couple of higher rolls and a couple of lower rolls. This isn't always the case, you might be unlucky and get all low rolls, on the other hand you might be lucky and get all high rolls. Typically, unless you're allowed to allocate the die rolls, it's unlikely that you'll get a specific type of character based off raw dice rolling.
What if we pushed things even further...instead of just attributes, we forced players to roll for other aspects of their character, then also made sure that mechanically these character elements had an impact on the ongoing story. High intelligence might give more skill proficiencies, or bonus languages, but now we also roll 3d6 to determine a character's social standing which has an impact on die rolls made in court settings.
Die rolls might determine race (3-5: Outcaste Race, 6-8: Wary Races from Afar, 9-10: Accepted Races from Afar, 11-12: Halfbreeds, 13-15: Impure Members of the Dominant Race, 16-18: Pure Members of the Dominant Race), where an inherent level of racism is fundamentally implied in the system, because those with high die rolls belong to the favoured races and correspondingly gain bonuses for simply belonging to such races.
Perhaps we're playing a typical misogynist setting where the impact of sexism is felt directly in the mechanisms of play (rather than just the flavour text or the misogyny of out-of-character interactions), but in this variant players are forced to randomly determine their character's gender (3-10: Female, 11-18: Male). You could go further and possibly incorporate sexuality as well as gender in some way.
I'm not saying to throw these ideas in and then disregard them during the course of play. A lot of games already do things like that, often creating elaborate backstories and dollhousing the character concepts, but this is a deliberate means to play with these identifying traits of the characters.
I'm not sure where I'm specifically going with this, whether it's a good idea or bad. It's just something I had to get out.