In a theoretical design, one of the best ways to handle this issue would be to have every place that is affected by an ability mod always be affected by two ability mods. Initiative is Dex and Int. Melee attacks are Str + Dex or Str + Con (depending on the weapon). Saves are explicitly called out as particular categories (preferably with clear, intuititve boundaries), with each one getting two stats--e.g. Versus Illusion as Wis and Int. If you aren't careful, you still end up short-changing Cha in such a system, but it can be managed.
Practically, though, I don't know a good way to handle the additional complexity. Averaging rather stinks in a system being built from scratch to support such an idea. Ideally, you'd want to simply add the two modifiers, but adjust the core math to compensate. That's fine, except that then you've just compounded the whole problem of ability score creep, that Next is working so hard to rein in. (That is, it doubles the range of possible variance in characters, though practically the extremes would be avoided. Not many characters would have dual 8s or lower, or dual 20s.) Worse, every single thing a character rolls against is now a derived value, also abandoning one of Next's nice features.
