One of my cousins enlisted in the Army, but due to medical reasons he is being discharged. How do I inform his mom that it'll get better?

I'm not understanding your question.

How do you inform his mom that WHAT will get better? Your cousin's medical condition? His life after being discharged?

And why should you be the one to tell her this in the first place? Is she asking? Is your cousin not communicating with her? Is she in some sort of panic over her son?

I could give some advice/opinion, but I'm lacking a bit on important details to go about it...and would rather not try to cover all bases and be excessively verbose.

It is not your problem to tell his mother anything. However, if you do want to say anything, don't rub in this person's handicap or disability. Sometimes it is better not to say nothing at all that to over insert yourself. Your cousin tried this out and since our Armed Forces is a voluntary one, then it is okay. As a military mom, I would rather you rub this in the wrong way to your Aunt. She will be okay with it and she will have a lot of support when she is ready. Thanks for your service and for your cousin's time in.