In real life, however, pretty much everything is mental and conveys just how much concentration or whatever belongs in the task.
Instead I propose a system that requires every competent character to have decent mental stats in some area. Even manual labor in the real world requires some level of intuitiveness or intelligence or awareness.
By doing this, the mental stats could be split up a bit more.
In this way, this new system won't have a highly generic 'intelligence' stat.
Here's the stats I've come up with. This is for basically any new system you guys may want to come up with. It follows the attributes of the brain outlined in 'Lumosity' - which is a website devoted to brain training games. The names have been re-named.
Alacrity (speed): Quickness of thought. Helps in situations where a character must think fast. One scenario could be reflex saves, initiative, reading the movement of another, or casting a spell more quickly than normal.
Awareness (attention): Awareness is the ability to take in a whole bunch of detail at once and process each and every piece of it. If a lot is happening at once, a character may not be able to focus on much unless they have a higher awareness score or whatever. This is also used for 'perception' based rolls, but only when something would get caught in plain view. Also helps a character multi-task - so a character can perform simultaneous actions if they have a high enough score or each tasks are well rehearsed enough.
Memory: More closely related to intelligence at its fundamental. If you have a high enough memory, you could take some abilities like 'photographic memory', or cast more spells per day, or cast from a scroll without having read the scroll for a while, etc.
Analysis (Problem Solving): Also closely related to intelligence at its fundamental. Good analysis means they're better at general Int checks or learning more complicated abilities (at the expense of requiring more concentration or whatever). An analytical mind is also more enduring.
Diversity (flexibility): The 'creativity' stat. A person with good diversity will be able to spot things that are hidden, and notice things that most people don't pay attention to. They also might be able to come up with new ideas due to their ability to think 'outside the box'. This might also be the charisma stat - a diverse mind can convince others of things like the fact they themselves are smart (usually by drawing upon a bunch of unique topics in just a short interval) or are able to put themselves in the other person's shoes.