Ok, forget most attributes, I want to talk about LUCK and only LUCK.
It's a fairly modest and straight-forward concept, which is offered with the details "as I would do it" for the sake of illustration. I want to hear about the overall concept. If it has legs, we can worry the details (or TESV will come out and I'll actually have to try to MAKE IT SO)
1. Luck starts at 0. No race, class, or birthsign is exempt.
2. Luck cannot be raised by conventional means (leveling, skill increases, etc)
3. Luck increases happen when you accomplish something specific, and never at any other time.
4. The relevant accomplishments should be outlined in books, rumors, notes, or other in-game bits of dialogue or text, but they should not be the most common occurances.
5. As a matter of design, getting to luck 50-70 should be easier in total than moving from 70 to 80.
So how would it work? Well, we're going to mentally retrofit this onto Oblivion as an illustration.
You come out of the sewers and you're down on your luck. Suppose you brave the bandits outside Vilverin and come across "Immortal Blood" (a skill book). Reading that book will not only increase your skill, but as your
first skill book, it increases your Luck to 1. The essentials of a TES skill book are that they seldom restrict themselves to teahcing a skill. There's almost certainly little details about Nirn in there that a crafty adventurer might use to gain that slight advantage (ie, exactly what Luck does). Now you swim south a ways and encounter the Wayshrine of Stendarr there. Praying there culd have two possible effects: it depends on whether "the designer" considers the chapels or the wayshrines "holier". If the Wayshrines are the "true holy sites", then you'd get a point of Luck for each one. Otherwise, you'd need to pray in the chapels (and KotN's Shrine of Kynareth for a 9th Luck point) However, praying to ALL of the divine wayshrines seems like an ideal "clear point of Luck" as well, so either way, this shrine grants some benefit.
On the other hand, a few hours later, you spot Vaermina's Shrine in the distance and check it out. Either you choose to serve her (and in my book, earning a little tolerance from a Daedric Prince is a sure way to help slant the world in your favor a little bit), or not. The latter is great for your morals and/or conscience, but as a consequence, you do have a major power that isn't pleased (as evidenced by the point of Luck you chose not to earn).
A while later, you happen on a rare volume of "King", a tale of Eslaf Erol. Having already chanced on Beggar, Thief, and Warrior, you've collected and read the entire series, and you understand Tamriel a bit better through their humor. One point of Luck (although it's reasonable to argue that longer series deserve it more).
In summary, Luck isn't something you can increase because you beat up enough mudcrabs under this proposal. It's something you have to either chance on, or otherwise act towards. Whether it is leveling skills you may not really use to reach Apprentice with them, or serving a Daedric Prince despite having some misgivings, or simply taking the time to track down rare books, or praying to the gods. You have choices and consequences: choose to be a god-hater, and you pass on divine favor. Choose to avoid the Daedric Princes for moral reasons, and you have no one to blame but yourself for the slight favor they might have granted. There's not COMPULSION here to power-game: it's not as if Luck is a crippling attribute. Each point of Luck is .2% missed potential (assuming the traditional Luck bonus to actions of Luck-50/10 = % change)
Meanwhile, it makes Luck stand out, and if anything, encourages the casual audience to really get into the world a bit more in search of ONE MORE point of luck