If there is a single mechanic thing that unites generations of pen & paper RPG players, it is the six abilities. Whether you are a grognard, or just came in with 5th Edition, in your free time, you're probably gonna roll some Strength, Constitution, Dexterity, Intelligence, Wisdom, and Charisma.
In a system I was putting together about three years ago, I tried to retool the ability scores into different categories. I tried everything. In the end, I came back around to the six we all know and love/hate. You are free to disagree, but I don't think we'll ever see a system for capturing your character's basic capabilities that is better than the six ability scores.
But once we have those six on lock-down, and we know we're going to generate them in some fashion within a range of 1-20ish, we as game designers should be squeezing as much out of those six numbers as we can. Let's talk about how to do that.
How about Basic Role-Playing's roll-under system? Zak S. talks about its elegance and drawbacks here, read up. It's worth noting, however (spoilers!) that he comes to the all-too reasonable conclusion: how you set target numbers, whether with stats, like the roll-under system, or as a separate scale you roll against, doesn't much change the feel of play.
So, stats as target numbers using roll-under? Eh, if you think it makes life easier, do it.
Now onto the real fun stuff.
Although the roll-under vs. hit DC dichotomy makes little difference, I am wholeheartedly a proponent of using the untranslated ability scores in your game. The great thing about roll-under is that it forces you to use your actual score during the game, and it appreciates every point of granularity that is there. Case and point: 4th Edition's "constitution score as 1st level HP" mechanic. I love that kinda stuff. It is easy to calculate, and makes each stat meaningful for all characters.
And I really do mean all the stats. Strength score? Well, how about that encumbrance system you were trying to fix a la here and here. Looking for everyone to actually use intelligence? Try here again. Over a Jovial Priest, there is a collection of stats-for-stuff house-rules here. There are a gazillion other ways to do it, but just make sure you do it.
Really. No more dump stats. What a waste.
Speaking of which, let's talk about modifiers. It is truly a sin that modifiers became the only thing used of the ability scores, when we had a perfectly good number lying around as the "score" itself. Having touched on that, though, we should address that even some modifiers were left out in the cold.
You should use all of them too. And not just with certain skill checks or saves. I mean your Charisma modifier should affect every social interaction you have. Your Wisdom modifier should protect your from all magic, or something! It doesn't have to be those rules, but it does have to be something.
Every number on the character sheet is a chance to further describe your character. If fighters in your game can point to their Charisma modifier of -3 and say "Eh, it doesn't really mean anything. I'd play exactly the same if I had a +1 modifier," then you, my friend, are wasting a creative opportunity.
Grognardia has a fantastic amount about old school use of ability scores and modifiers here, here, and many other heres. He discusses ability scores on a much different level than I am currently doing, but it is useful as all hell regarding game design.
I am generally not a fan of strict game balance. But I am a fan of strict ability balance. If characters with high Strength do significantly better than characters with high Wisdom in essentially all aspects of your game, you are bottle-necking your players.
You solution is one of two things: either you add to your game to make the under-utilized stats more useful (my personal choice), or you subtract from your game to remove that hangnail-esque mechanic.
If only sorcerers use Charisma in your game, and they only use it for magic, take it out. Or conjoin it with Wisdom and make it "Personality" or something. Seriously. I don't tie classes to abilities in my games, but it is a very common rule. If you are going down that path, don't screw over certain classes by tying them to crappy abilities. Start by making each ability worthwhile in its own right, and then tie it to the class that makes the most sense.
There will be plenty more on this to come, but I think this is enough for today.