I'm not Rioter, but from my understanding power budget is what your champion can do, and how good they get to be at it. The best way to look at it would be in terms of opportunity costs: in the end, you want your champion to do well in certain situations, less well in others and have a certain level of appreciable power all the way, so if you start from a "balanced" kit, if you want to add something you're going to have to remove something else, and vice versa.
One important part of this is that reliability and low counterplay themselves factor into this power budget, so when low-counterplay champions get Olafed it's generally to account for the unhealthiness of their kit.
is an example of this, because so far he's been kept fairly weak overall, but can only work in that state because of his super-safe poke pattern, which involves little interaction with his opponents. His power budget is getting (chrono)shifted away from things like safety, low-counterplay poke and perma-slows, and more into stuff like stronger and more powerful CC.
If you were to take
as an example, a lot of his power budget is going towards the confusion and frustration created by his playstyle, along with his oppressive laning phase, which is why he has horrible mana problems and terrible late-game scaling (the latter's also because his ult lets him copy the marksman, which is super-strong in the late game). If he were made less oppressive and confusing, he could afford to have a lot more power elsewhere (on his ghouls, for example).
Picking an example that doesn't hinge on counterplay:
has a pretty healthy kit, but is still more of a burst mage than a proper plant mage, as her EQR rotation is often enough to 100-0 squishies on its own. That's not necessarily a bad thing, but it also means her plants can't be stronger. If you were to remove her plants entirely, Zyra could have much stronger abilities, and if you were to reduce the damage/CC on the non-plant parts of her kit, she could have much stronger plants. Either way, she'd likely be balanced and healthy, and would perhaps fit a stronger niche, though I'd personally rather go for option 2 (making her more plant-based at the cost of her other abilities).