My brain immediately goes to Final Fantasy for this question. It is my core reference point for what a class system ought to look like. And if we map FF1's original six classes onto the D&D ability scores, it looks like this:
Strength — Fighter
Dexterity — Thief
Constitution — Black Belt
Intelligence — Black Mage
Wisdom — White Mage
Charisma — Red Mage
Or, in D&D terms: Fighter, Thief, Monk, Magic-user, Cleric, and Bard. And it does, quite naturally I think, stand to reason that the monk and the bard ought to be considered "the" iconic Con and Cha classes respectively. (At the very least, they're the only two classes which are not sub-classes of the F/MU/C/T quartet in AD&D 1st edition! Pre-cavalier, anyhow.)
When I choose to depart from D&D orthodoxy, however, it mostly centers around spellcasters and their ability scores for me. In some cases, I like for mages to be the Charisma class; in others, I prefer clerics to be the Charisma class (which makes a great deal of sense to me if only for the etymology of "Charisma").
For example, my current Basic D&D house rules use the following six classes: Fighter, Mage, Cleric, Thief, Adept (a psionic monk), and Ranger (which I portray as much more gishy and loremastery than the traditional warrior-hunter-tracker — my rangers are basically also bards flavor-wise, though their primary function in my games is to replace the B/X elf fighter/magic-user class with a less obviously overpowered fighter/mage/cleric mix). Things don't quite map one-to-one (because I intentionally burden Adepts with a dependency on two key ability scores and Rangers with four), but this is what my current setup looks like —
Fighter — Strength
Mage — Intelligence
Cleric — Charisma
Thief — Dexterity
Adept — Wisdom (secondary dependency on Con)
Ranger — Constitution (secondary dependencies on Str, Int, and Cha)
I like this, because it makes Intelligence the key score for arcane magic, Wisdom the key score for psionics (but you can't be an effective psionicist unless you have both a healthy mind and a healthy body), and Xάρισμα the key ability score for divine magic.
When I set games in more modern-day or futuristic milieux, meanwhile, I tend to shuffle things around so that Intelligence is the key ability score associated with artifice, science, and technology. For example: in the next edition of my steampunk OSR game, Engines & Empires, I plan to have the game's six classes map to the ability scores as follows:
Fighter — Strength
Thief — Dexterity
Ranger — Constitution
Engineer — Intelligence
Scholar — Wisdom
Mage — Charisma
In this game, the fighter, thief, and engineer are all fairly straightforward and self-explanatory. The core operation of the mage class involves a lot of in-universe ritual magic and spirit-binding, so it makes sense to me that Charisma would suit as the class's prime requisite (every time a mage does something magical, they're either bargaining with a spirit or bending a supernatural force to their will; and anyway, if we take Gandalf to be the iconic fantasy Wizard, it just makes sense for sheer presence to be the key quality of a good mage). The ranger and the scholar, meanwhile, are the game's hybrid classes, with the scholar serving as a kind of cleric- or bard-flavored fighter/mage (it essentially replaces the cleric, but with an occultist/Van Helsing vibe — and if there's a point to Van Helsing in the original Dracula, it's his wisdom, the fact that he's both on the cutting edge of modern science and philosophy and has a healthy respect for older, less rationalistic worldviews); and the ranger being an outdoorsy explorer/pioneer type who serves as a fighter/thief/engineer mix (with a focus on crafting potions, drugs, and chemicals, Witcher style). In a milieu like this, Constitution makes sense as the key ability score for a class that gets flavored as a rough-and-tough "fantasy cowboy" archetype.