Player approached me with an idea that I figured I'd ask about. 

Do you forsee anything breaking if I let a player, as part of chargen only not on the fly, swap one of their class' secondary attributes around?

Example would be someone wanting to play a Monk who uses Cha instead of Wis for their ki pool and unarmored defense, or a 'studious' Ranger who has Int based instead of Wisdom based magic.

Presumably you realize that this will have impacts on other things than just the Ki pool or magic spells (for example, all INT-based rolls will be affected for the Ranger).

I probably would not allow it personally, but I don't see why it would majorly break anything.

As BioWizard said, it probably won’t break anything but is there a good reason why they would like to do so?

if, in the example of the monk using CHA do they plan on then multiclassing into a CHA based class then I might not let them if they are just trying to game the system.  

but if they had a good reason from a story, background, RP perspective then go for it. 

I would be tempted to give a flat out no WotC did a lot of work balancing things and a change as radical as this could lead to issues

If I did allow it  I would be wary, in principle a Cha based ranger would be gain on int based skills and lose on wis based proficiencies so these should roughly balance out. I would find a ranger with very low perception somewhat odd and that would need to be written into the background.

However I would be suspicious that the player has something up up his sleeve that will make the character OP. Ask how he plans to develop his character, I would ask him to commit to what options he would take at the next level at least a session or two before you actually level up so you can see whetrher their are any potential issues.

A might also rule out the character multiclassing as this is likely to be the biggest cause of balance issues.

As I have said, I would not allow it. But if I did, I agree, no multiclassing. A player asking to be, say, an INT-based ranger smells of someone trying to max-out a ranger/wizard build or something along those lines. If you want to be a INT-based fightery class, subclasses already exist that let you do this (Eldritch Knight, Bladesinger).  I'd say do what exists in the rules first, before trying to homebrew something that may not work as intended. Same with the cha-based monk -- is the player just trying to get a ridiculously maxed out monk/sorcerer or monk/bard?

However, that is my take... I think that keeping it as a single-class character would probably work and likely wouldn't break stuff.

Quote from BioWizard >>

I think that keeping it as a single-class character would probably work and likely wouldn't break stuff.

This pretty much sums it up for me. Whilst there shouldn’t be anything too bad with a pure monk using Cha instead of Wis, the issue comes when multiclassing as Bards, Sorcerers, and Warlocks all benefit greatly if you multiclass in to those.