Yes and no
Whether or not it breaks the balance of the game largely depends on how you execute the solution, especially since you're in a homebrew setting. Just providing a flat increase on a character's spell save dc is naturally going to introduce power creep beyond what the developers intended and would throw off the balance that you are most likely accustomed to. However, balance is affected by many elements of the game, and by altering multiple elements at once you can still change it without breaking it. That is to say that you can preserve the balance by creating some sort of counter-balance to make up for the increased dc.
Essentially what you're talking about is specializing into a specific play style. It sounds like what your player is looking to specialize in is a greater consistency and dependability with their illusion-based spells. However, in order to avoid power creep in these kinds of situations, specialization must come with trade-offs. Groups I've run with have dealt with this in varying contexts from traits to class abilities to spells. This almost always requires some degree of tuning after implementation, but usually serves as a very stable platform for defining a play style.
An example specific to your situation that comes to mind would be giving them an item, lets say a circlet, that increases their spell save dc by two, but also shortens the duration of any illusion spells they cast. You are then giving your player the option to trade potency for consistency. Also, don't forget to justify this in the narrative!! This is D&D after all.
Here's a quick mockup of what that would look like... (feel free to use this if you'd like)
Crown of Thitus, God of Deceit
Attunement slots: 1
- The spell save dc for any illusion or enchantment spell you is increased by 2
- The duration of any illusion or enchantment spell you cast in halved
- Any illusion or enchantment spell you cast immediately ends after 1 hour
Thitus was once a mortal man, but was born with a dangerously cunning and mischievous soul. He had a natural talent for the arcane and reveled in conjuring grand illusions or bewitching others with fierce enchantments. But Thitus could never be satisfied. He easily grew bored, losing interest in his games as quickly as he created them. Yet his power continued to grow even as his interest in humanity faded. One day he decided to seek audience with the gods, promising himself that this would be his final game. And so called upon his cunning and illusions and enchantments one last time, and he fooled the gods into granting him ascension. He became one of them. Among their ranks he was established as the patron of tricksters and lies. Now he sits upon his illusory throne, watching humanity, waiting for someone worthy to take his place.
The main problem I've seen with this solution is the Swiss Army Knife effect, wherein a player or party builds a massive arsenal of magic tools like this that they hot swap between to gain the edge in every fight. This may not be a problem for you depending on you and your players' styles, but it is something to be cautious of.
Another (less easy) Solution:
You could give your player an item that simply provides a flat increase to spell save dc without any drawbacks, and instead implement a counter-balance by manipulating individual encounters. This is far more difficult as each encounter is going to be different and call for unique changes, and you must also take into account every other player in the party. For example, if you create a monster which deals damage to a random creature within thirty feet of it, this could harm not only the illusionist but also any other party member within thirty feet of it. There are a lot of moving pieces in this kind of solution, and though it gives you more control over each situation, it is much, much harder and more time consuming.