If you were designing your perfect game system, what attributes (strength, intelligence, etc.) would you include?
It doesn't really matter what you call them.
(There are also fine games without ANY attributes, that is without
stats that any character of a certain type will have at a certain value.)
If you want to have such stats make sure that either you do not have more than six or sort them in smaller packages, like WoD does. (Magic number seven.)
Furthermore make sure that each attribute has a unique mechanical effect that everybody wants. That way it sucks to be bad in any attribute. (D&D is quite bad in this respect, although many other games are even worse.) If you cannot come up with six unique mechanical effects, you do not want to have six attributes.
If you use the traditional split into turn based combat and free play, it's useful to give every attribute some hard use in both areas. (7th Sea is a nice example.)
The result of this considerations is that you are better of thinking about mechanical effects first, then turn them into attributes, then label them appropriately. Not the other way round.
How would you rate the abilities and how would you generate scores?
That depends very much on the game you want to create. Random creations often has the idea of a less serious game associated with it, but that doesn't have to be the case. (See Greg Stolze's Reign for a mechanic that creates a whole character by rolling 11d10.)