(I'm going to ignore the argument over disability/abnormal/disorder/illness/diseased/etc because I really don't think that's answering or talking about what the OP has asked)

To me, having a mental illness (I am using OP's terminolgy here) means that I have challenges to face in how my mind processes information and emotions, and challenges with my behaviour. These challenges are often more extreme/pervasive/abnormal (because yes, some of those challenges ARE worrying or undesirable to me!) than what the average person faces. The challenges will cause my life difficulty because I will not always be up to functioning the way that I would normally. Sometimes I enjoy the changes (I have bipolar 2, so I'm referring to when I'm hypomanic) because there can be benefits - when I'm hypomanic I feel a lot more confident in myself and am often much more productive.

Really, to me, it just means that my brain works differently as the paths in it didn't all form the way that they are ideally "meant" to (I use that term because yes, my brain is wired differently than human brains are ideally meant to - loads of people have various miswires...), so I have to try and rewire my own brain using various methods - which ones are up to me. They happened to be wired in a way that most matches the bipolar 2 diagnosis. If my brain was wired averagely, then I would not be diagnosed with any disorder as I would not have one.

And that is what having a mental illness means to me. It will mean something different to different people.