Here are some game ideas that are probably stupid ideas but I never claimed to be a genius.
D&D types: experience points are hit points
Dump hit points; instead, all damage is taken from experience points, and if you lose enough experience points you can also lose levels. For example, a third level Lamentations of the Flame Princess fighter who drops to 3999 experience points becomes a second level fighter. There is no healing. To recover from your injuries, you have to go out and get more experience points.
To avoid starting with characters that are already dead, consider starting at second level, or maybe rolling hit points as normal to determine starting experience.
Call of Cthulhu: professions instead of skills
In the latter half of 2016 I was running a Dracula Dossier game using a variant of the Call of Cthulhu rules, and over the past couple of weeks I've been playing in a Call of Cthulhu game using a variant of the newish Delta Green rules; in both games we have put together a bespoke skill list rather than using the default skills, and that got me thinking about simplifying the process. Then I thought about how skills in 13th Age and Warhammer Fantasy Roleplay work, about the weird random skills CoC non-player-characters sometimes get, and all of that led me to this.
All characters start with 55 SAN and 11 hit points. They then get 120 build points to spend on professions, hobbies, hit points, and Sanity. Professions are loose descriptions of what the character does, but not too loose; "professor of archaeology" is good, but "academic" is rubbish. Instead of having a list of skills, someone with "professor of archaeology" can do things that a professor of archaeology could do; they'll be good at history and geography, they probably know a bit about digging holes, and may know a bit about architecture and languages, but they are probably not going to know how to fly an F-15 Eagle.
That's what hobbies are for. These aren't the character's main profession but side interests, or previous jobs, and work in the same way. "Former soldier" is as valid as "member of snooker club" or "bakes on weekends".
There's going to be a bit of back-and-forth between player and GM about what is reasonable for a character to be able to do, but as long as everyone is sensible it should be fine.
I suggest limiting the main profession to a maximum of 80, and single hobbies to a maximum of 40. Starting SAN can be increased to a maximum of 99 by spending points at a one-to-one rate, and hit points can be increased to a maximum of 18 by exchanging five build points per hit point. If you want, you can sacrifice SAN and hit points, at the same rate, to gain build points; minimums are 3 hit points and 5 SAN.
(I did consider dropping hit points and SAN and having all damage -- physical or psychological -- apply to the character's hobby or profession scores, but perhaps that's a bit too abstract.)
Any: don't roll to hit, go straight to damage
I thought of this in terms of D&D type games but it could work with any game in which there are separate die rolls for hitting and damage. Instead of rolling to hit, just roll damage. Simple. It does mean that every attack hits but it also eliminates the disappointing naffness of rolling a 19 to hit then rolling 2 damage.
It could penalise some characters in some systems; for example, a fighter in LotFP loses one of their key advantages, the increase in attack bonus. If that sort of thing poses a problem, perhaps add the attack bonus to the damage roll, although that may be too much of an overcompensation in some systems.
Now I've put those stupid ideas into words they will perhaps exit my brain and leave space for something more useful. I haven't tested any of them-- I probably wouldn't have published them if I had -- so instead I release them to wreak havoc on someone else's game. Sorry.