Would this idea be too sensitive / too much information / generally a bad idea?
For example, I have a friend who is doing her computational biology PhD at a strong U.S. program, and her main research is about the illness of one of her family members, whom she unfortunately lost recently.
I have also heard of specialist (medical) doctors who got into their field because they suffered from the very illness that they are now trained to diagnose and treat.
For me, at a much lower level, at the MS Thesis level, I have a chance to work with a very well-known applied mathematician, who has joint appointments at our math dept and medical school. But, I don't have a clue what project I would like to initiate with him, other than about something that I have dealt with personally and medically -- and this topic could likely be within his research domain. Of course, I would be naturally very passionate about working in this area. I am always fascinated, when I see my own team of doctors twice a year to have discussions about my health.
If I could do a mathematical project about my own health issue, do you think that this is ... TMI? A classic "beginner's mistake" perhaps? Or, is it pretty common for applied mathematicians to dedicate some (or all) of their research to issues that they are intimately familiar with?
I do not have experience with specifically applied maths, but in general, there is no problem at all with conducting research in a direction that you have some personal connection with. One could argue that many applied research projects are driven by concrete needs, and I fail to see why those would have to be needs of other people rather than a researcher's own.
That being said, you will need to remain cautious of your own bias in your endeavours. As a researcher, it will be expected of you to work on the topic from a neutral, scientific, "big picture" perspective, which is easy to lose if you are at the same time also part of this "picture".
Let me provide one example from my own research. I conduct research on software engineering. Many of our best PhD students have worked as software developers prior to starting their PhD, so in a way they are also part of the same demographics that they research. What can now happen with inexperienced students is that (s)he will, for instance, interview developers with regard to a specific problem, process, or practice, and, during the analysis phase, discard statements that go against their own prior experience as "obviously wrong" or "bad". In that way, a student may overemphasize results that resonate with her/his own prior experience, and discard results that don't, essentially guaranteeing that the results are in line with what the student expected going into the study based on her/his own experience.
You will need to make sure that you do not run into the same trap - don't assume that everybody experiences your health issues the same way, that you are some sort of gold standard of this specific condition, or that your specific variation is more important or relevant than others. However, these are definitely issues that a good advisor can help you with.