This is just a little idea I had for Lamentations of the Flame Princess. It could more-or-less be used the same way in most versions of D&D. I haven't playtested it yet, but I wanted to jot it down.
Ignore the primary ability scores and just use the bonuses. Ability scores now simply cover a continuum from -3 to +3. All characters start at first level with two ability scores at +3 (equivalent to a score of 18), three scores at 0 (equivalent to the 9-12 ranges), and one score at -3 (equivalent to a score of 3). Use a different array of ability scores if you prefer. You can have players choose which abilities get which scores, or you can determine it randomly in true old-school fashion.
If you're using the Playtest Document 0.1 rules, ability scores would go from -2 to +2 instead, with these numbers being shorthand for the five tiers of ability score bonuses, like how Con makes HP goes from d4 at the lowest to d12 at the highest.
If you want to start PCs at level 0 instead of 1, you could make one of the +3 scores another 0, and let them bump one of the 0 scores up to +3 if they manage to claw their way to level 1. Or you could start level 0 characters with scores of 0 across the board and cruelly make them pick a score to drop to -3 along with two scores to raise to +3 if they manage to level up. This could reflect some kind of injury or trauma they suffered during their first adventure, like what happened to the Flame Princess' leg and fingers, or massive sanity loss, or some jacked-up facial disfiguring.
Whatever the case, whenever a PC reaches a new level (2 or higher), they get to pick one ability score bonus to increase by one point. At level 2, a PC could make that -3 into a -2, or make one of those 0 scores into a +1. Likewise, if a PC loses a level, they have to pick an ability score to decrease by one step, or the DM could make them determine an ability score to decrease randomly. If my math is right, in the current LotFP rules, all ability scores could be maxed out by level 16, and in the Playtest rules they would be maxed out by level 11. If this seems too nice, keep in mind that with LotFP's steep experience point requirements (especially in the Playtest rules), characters usually don't even survive to level 11. If they do, I think they've earned some better ability scores!
If you're playing an adventure with monsters/traps/items that drain ability scores directly, consider limiting any single instance of ability score drain to one point, since this new range of scores only goes from -3 to +3 (or -2 to +2) instead of 3-18. In this setup, losing one point would already often be the equivalent of losing several points in the vanilla rules. Ability score drain could be used to offset the constant increase per level if the DM thinks what I've proposed is too rapid, although characters should still generally have a fair chance to avoid ability drain, of course.
Honestly, one of my least-favorite aspects of old-school D&D and games derived from it is the concept of ability scores that do not increase with a character's level. If I wanted to explore the concept of one's capabilities being mostly determined by one's birth/station/other immutable factors through the mechanism of almost-unchanging PC statistics and abilities, I probably would not choose to do so in a fantasy adventure game that holds as one of its main draws the ability to become more powerful and competent (and sometimes more heroic) over time. I get that LotFP is kind of a pessimistic, horrific take on D&D, but the lack of non-magical ability progression still feels more disappointing than fulfilling. I mean, if the only reason you're keeping levels at all is to use the promise of experience points as a carrot to lure players into awful places that they're better off not exploring and would enter under absolutely no other circumstances, a promise that often leads to disappointment due to the the game's difficulty and themes of pessimism and horror, why not just get rid of levels altogether? Why half-ass things by cutting down character advancement to almost nothing? I'm sure a group that really wants to play a horror version of D&D could brainstorm together and quickly come up with some other justifications for adventuring. People go on dangerous adventures in Call of Cthulhu, after all, and those folks are usually pasty academics and assorted nobodies, not hardy adventurers.
Well, I guess keeping a leveling system helps with compatibility between LotFP and other games, which helps Mr. Raggi make money, and I absolutely can't blame him for that, especially since I fucking love his work. But that's not a satisfying enough reason to keep things working the same way at my table.
I also think it's a little silly that PCs can't get stronger through exercise, or wiser through their experiences over time, or what have you. (See what I did there?)
Maybe I just feel this way because I'm used to the way RPGs are done in video games, where there usually isn't a statistic displayed in a menu that you can't improve through use or experience. I just don't think mostly-static ability scores make a lot of sense, from the standpoint of game design, the game's themes, or "fun" (there's that word Mr. Raggi doesn't like). I could be missing the point here, but I think more is gained than lost by allowing ability score improvements with leveling up. Besides, I prefer to have leveling up be a big deal, and I think increasing ability scores help with that.
That's just my two copper pieces on the subject.