The six core statistics of STR, CON, DEX, INT, WIS, CHR (arranged this way because they roughly divide into the "physical" stats and the "non-physical" stats) work pretty well and are responsible for the majority of a character's ability description: Can he break down a door? Can she drink Dwarven Spirits? Can he dodge an arrow? Can he out-think a dragon? Does she know better than to try to out-think a dragon? Can he convince others he can out-think a dragon?
Unearthed Arcana gave us the idea of Comeliness (CMS), and a lot of Narrative DMs really like the idea. Comeliness is sheer, physical beauty. Since The Lord of the Rings came out as movies, I've been able to use this metaphor: Comeliness is what Galadriel spills all over the place in The Fellowship of the Ring. When people see her, before she opens her mouth or pokes in their head, they go "aderp, derp, dubba derp" and fall over themselves. When she then opens her mouth, she basically turns everyone to jelly. That's because Charisma modifies Comeliness. Assuming the people hearing you understand you, a high Charisma raises CMS, a low CHR lowers it. If you've ever walked up to a hotty in a bar and the dreck pouring out of its mouth sent you fleeing, you've experienced this.
I have my players roll straight 3d6 for CMS, and recently added that they may discard the first roll for the second, but if they had a 9, and reroll and get a 3, they are stuck with that 3 (probably.) This is because CMS is completely luck of the draw...the world is full of pretty villains and ugly heroes, and relying on physical beauty to tell you anything about anyone should always be a mistake. A charisma-based character with a low CMS can always take steps to hide his/her ugliness.
The second non-standard statistic I use is LUCK. Players roll percentiles three times and choose the best out of three. In a normal gaming situation, luck is never rolled. LUCK, however, allows a player to feel some modicum of control in an uncontrollable situation. If, for example, a character has just been tossed from a tower on a flying cloud castle, and failed any saves or stat check, a luck roll can give the player one last chance to live...or at least let them feel they could've stopped the inevitable death. How much of a negative is applied to the LUCK roll depends on the situation. Luck can also be used as a warning when a player wants to do something probably statistically impossible:
Player (with 18 DEX): "Do I think I could climb safely down the tower?" (which is slick with flammable oil, and it is a lightning storm outside.)
DM: "Probably not."
Player: "I guess I have no other way out, though...I prepare to try to climb down the tower-"
DM: "You gather your rope and your spikes and make for the window. What is your LUCK?"
Player: "I look for another way out."
Luck can be used to give a character one last chance after a failed saving throw- instead of disintegrating him and blowing him away, maybe just his arms and legs are disintegrated, or instead of the death spell slaying the Paladin outright, maybe his +5 Holy Avenger took the attack, shattering into a thousand pieces, but sparing his life. Any situation where LUCK saves a character from complete obliteration, there should still be something nearly as bad happening to the character.
Another use for LUCK is when a non-cleric character mentions a deity, or a cleric character mentions a deity other than his own. With a high enough (or low enough) LUCK roll, the deity might happen to be looking that way. This is possible with high level NPCs as well, and should be understood as a somewhat cumulative concept. One day, the guy who keeps going on about the Hand of Vecna is going to have a Hand of Vecna appear, and it will be attached to a Wrist of Vecna, and an Arm of Vecna and, in fact, the complete Vecna. When the Vecna of Vecna attacks the 3rd level party...much LUCK will be rolled.
[I personally keep a running tab on piety. A cleric going on about his god a lot (unless doing so is against his religion) is much more likely to get special assistance from his god than one who only calls on him for spells and turning undead. I don't consider this the same as LUCK, or even related.]