we cant diagnose your friend only their treatment provider can do that so the part of your post about your friend I am going to pass on by, but your question in general is there a mental disorder that includes when someone wants to have a disease or mental disorder.... yes there is such a thing. its called

Fictitious Disorder Imposed on Self. this actually includes many different labels. short version its any time a person purposely sets out to cause their self to have a problem that they wouldnt normally have.

here is where you can read about it... Factitious Disorders | Cleveland Clinic

please remember that there is more to diagnosing a person then reading something online, only your friends doctors can diagnose and treat your friend.