(Another mud-sprung idea, off some discussion in how to change the training rules in S7S)
Venturer! is a system for old-school dungeon exploration and smiting. Characters are intended to be quick to create, quick to play, and (unless you're lucky) quick to die. The game works as follows:
The first thing to do is roll your stats. Venturer! characters have six stats: Might, Athletics, Craft, Lore, Diplomacy, and Perception. Might is for strong stuff, Athletics is for everything else physical not covered by other stats, Craft is for clever stuff like lock-picking and slight of hand and solving riddles and doing math, Lore is for history and knowing which kind of plants make you itchy and everything else mental not covered by other stats, Diplomacy is for charming people or seducing them or bluffing them or intimidating them, and Perception is for noticing things. For each stat, roll 1d6 and write down that value. Stats range from 1 to 20, so you're not going to be great at anything, but this may be enough to decide on a background. If not, you don't need a background. But you do need a name! Write that down. And a gender, write one of those down too. The last thing is to pick some equipment: you get three pieces. A warrior might want a sword, a shield, and some chain mail, and a wizard might want a staff, a crystal ball, and a fireball spell. Ok, that's all, you're ready to go.
Play works like you'd expect — you wander around town until the GM has one of the NPCs tell you about an old abandoned temple or mine or something, filled with treasure and loot, and then you say you're going out to explore. Then you wander around until you have to make some rolls.
Rolls go like this: the GM assigns a difficulty (like 10) and a roll modifier (like +0). You roll 1d20 and add your relevant stat and the roll modifier. If it's equal to or greater than the difficulty, you succeed. If it's 17+ or you rolled a natural 20, not only do you succeed (always, regardless of the difficulty), you raise your stat by 1 point. If it's 3- or you rolled a natural 1, not only do you fail (always, regardless of the difficulty), you lower your stat by 1 point. If your stat goes below 1, you're dead. If it goes above 20, then you immediately retire and get a cushy post at some university teaching whatever stat it is (even if you're in the middle of the dungeon — it's all part of the tenure system).
The difference between the difficulty and the roll modifier is that the difficulty just affects whether you succeed or fail, but the roll modifier does that and also affects your chance of your stat going up or down. Oh, and the deal with equipment is, if you're using a piece of equipment, the difficulty is 2 less.
Oh, also, the GM never rolls any dice in this system, and the players get to choose what stat and equipment they use on any roll (but the GM can veto it, because we are old school). So play is like:
GM: Ok, you're in a room and there's a goblin here. He snarls and charges at you.
Player: I smite him in half with my axe! That's using Might and my Axe.
GM: Ok, the difficulty is 12, roll modifier of +0, so you need to roll a 6+ since your Might is 4 and you're using your Axe for a -2 difficulty.
Player: A 12! I cleave him in twain!
GM: Good work! But now his buddy jumps on your back and starts gnawing on your neck.
Player: I roll on the ground to crush him beneath me — that's Athletics.
GM: Ok, difficulty of 10, roll modifier of -2. You have an Athletics of 2, so you need to roll a 10+.
Player: Oops, I rolled a 3.
GM: With the -2 roll modifier, that's a total of 3, so your Athletics drops to 1. The goblin is totally chewing your ear off, what are you going to do?
You could have races, where they roll different dice for the stats (Dwarves roll 1d8 for Might and Lore, and 1d4 for Diplomacy and Athletics; Elves roll 1d8 for Craft and Lore and Perception and 1d4 for Might (if this seems unbalanced, blame Tolkien, man)). You could have different kinds of equipment, that give larger or smaller bonuses. You could have monsters that do automatic damage to one of your stats. You could have additional penalties from failing a roll, like when that guy's ear got chewed off the GM could say he loses a point of Perception too. You could do a lot of things.