It may be right under my nose, but I can't seem to find it anywhere!
What are the DCs for various monster abilities derived from? For example, a dragon's breath weapon indicates a DC, but, which conditions alter that value?
First recognize the priority is on the DC being suitable for the monster's level, with the stats it's derived from being retroactively altered. Usually. If built like a PC, it's basic stats building up to final stats & DCs. If built like most monsters, it's final stats & DCs being determined first, and working downward. Some of the AP's NPCs have wonky, superhuman stats because of this style! Paizo said they'll try to avoid that in the future.
So you can base the DC on the monster equivalent of class DC.
I just glanced at the Adult White Dragon & it appears to match this.
10 base + 10 for level + 7 for Str + 2 for Trained = DC 29.
One caveat is that AoE effects seem to have a -2 ingrained, so the dragon might well be +4 for Expert & -2 for AoE.
Or you can go to the monster-building charts, find the appropriate DC, then adjust the stats to match. If perhaps you didn't want the monster's "class stat" to be so high, you might make it an "Expert" at being a monster to get the DC up to norms that way.
One example, though for a gaze attack, is the Medusa.
10 base + 7 for level + X for stat + 2 for Trained = DC 25.
So X should be 6, but the Medusa doesn't have a stat of 6, so the DC could either be from Con & she's an Expert or it could be from Cha & she's a Master. Given that it's an arcane aura, I'd lean Charisma even though Master seems a stretch. Except her poison, which suggests Con, is also DC 25 so in the end I'd settle on that as her "class stat".
So I can tell you that Enfeeblement (Str effect) seems to be the condition that would alter the dragon's breath weapon's DC, though it's not outlined clearly. You pretty much have to reverse engineer if your PCs are regularly applying conditions tied to stats. The other conditions that might effect a DC should likely alter all DCs anyway.
The monster-building charts Castilliano refers to are in the Gamemastery Guide Monster and Hazard Creation Guide PDF (free).
There is no explicit guideline for setting the DC of non-spell abilities presented in the monster creation rules. It's been easy enough to surmise the intended guidelines by looking at the Bestiary's numbers, but hopefully the actual print edition of the gamemastery guide corrects the oversight. It was brought up on a thread for catching bugs in the monster creation rules, hopefully the developers saw it before the book had to go to the printer.