STR: Warrior - A sort of Fighter/Barbarian combo. Manuevers would be a core element of this class.
DEX: Hunter - Rogue/Ranger combo, with choices letting you determine exactly what you were.
CON: Psion - Using a new rules set, not slot-style casting, and avoiding anything that didn't have a check/variable result.
INT: Sorcerer - This would be the main "caster", and you could be a Wizard-type via subclasses/choices, or say, a DND Next playtest-style Sorcerer or a Warlock or the like.
WIS: Priest - Cleric/Druid combo but broadened out further to include things like Warden.
CHA: Wayfarer (Placeholder name) - I feel like you could probably fold aspects of Bard and Paladin into the same class pretty easily, and have a class whose main deal was inspiration and bravery, for good or ill, and where through choices and subclasses, you could come out with anything from a Lore Bard to a better Swords Bard to a Devotion Paladin to an Ancient Paladin, more or less. An alternative design would be some support elements plus the ability to pick low-end stuff from all the classes.
Also, I'd be making some big changes to spells. We wouldn't have any bigass level 1-9 spell lists/charts - spells probably finish at level 5, and most spells which are absolutes would require some kind of check and variable results. On the flip side I think all caster classes should probably refresh some spells on a Short Rest-type basis. Healing would absolutely NOT be limited to certain casters either - if the INT caster wants healing/protection/res spells, they can have them.