Should universities give extra time or related accommodations for students with mental illnesses?
In answering this question, I think that it is important to consider the distinction between mental illness and learning disabilities.  While there is a spectrum of different and complex effects, for the purposes of making policies it may be useful to consider these two "prototypical" classes of challenge:


Many specific learning disabilities, such as dyslexia or speech impairment, have a rather focused affect on perception and action, rather than more generally impairing cognition.  In many cases, relatively simple interventions (such as giving a dyslexic student more time) can have a big impact on educational success by mitigating the impact of the disability.  I think of this as little different than the intervention of glasses in mitigating the impact of my poor eyesight.
Mental illness such as depression or bipolar disorder, on the other hand, is much more subtle and difficult to deal with, since it strikes at "core" aspects of an individual such as motivation, interpretation of the world, and outlook on life.  Even when an intervention is plausible, the ethical boundaries are much less clear, particularly given the difficulty in balancing the need for consent against the likely harms from a possibly distorted mental state.


For an educational institution, then, the approaches to supporting students with mental illness and specific learning disabilities are likely to want to be very different, and the type of support that is needed is likely to be different for different types of challenges.  Giving a dyslexic student more time on an exam makes a lot of sense; it's like letting a student who is hard of hearing sit up in the front of the lecture hall.  Giving a highly depressed individual extra time on an exam is not likely to be a meaningful intervention; letting them take time off without penalty and return in another semester with a clean slate is more likely to be effective.

A good institution should then try to figure out the needs of each student, and how far they can reasonably go to satisfy them given the institution's resource constraints.