The case for mentioning:
A chronic illness forces a level of maturity and independence that cannot be faked. Depending on the illness, the kid has likely successfully juggled not only the standard high school challenges (learning, studying, taking tests and writing papers, volunteering, performing, being a friend/teammate/etc.), but also a whole series of other complex tasks: managing medications and prescriptions, calculating doses, actively planning ahead for things like sports, extracurriculars, field trips, etc. They've likely had to navigate a labyrinthine medical system, actively communicate their needs to doctors, coaches, teachers, friends, and school administrators.
If they've managed to do all that while remaining a strong student and a positive, personable human being, then they've proved something really, really important about themselves: that they can multitask, plan/ prioritize, deal with setbacks, self-advocate, self-regulate, and navigate complexity.
The challenge is how an applicant can communicate this without seeming like they're complaining, or as if they're struggling in a way that makes them less likely to succeed (or are more likely to be a drain on resources). Additional Information is a great place for this. Keep it brief and positive. Focus on the growth that's already occurred, rather than on the struggle.