Sometimes people add a name for a sick person. Can you just change it to an entirely different name? I know of a case where they kept the first name and changed the second name - originally the name was Menachem Mendel and they changed it to Menachem Chaim. I do not know if a Rabbi was asked and I am curious if there is any source for doing so.
Indeed, that is how the minhag began. The practice of changing one's name to an entirely new and unrelated name is attested in many places relevant to gitin because of the precision in names necessary there. (citation needed)
Incidentally, it seems that the minhag was drawing upon the auspiciousness of the name changing and not the meaning of the new name, so it was just as likely for בנימין to turn into ראובן as חיים.
I don't remember my source for this, but I recall learning that they used to completely change someone's name (not just his second name). Our adding a name is a modified version of that.
The Gemara (Meseches Rosh Hashanah 16b) says that it is a change of a name which is a cure for an illness. It doesn't specify whether you change your name completely or you just add another name to yours, I would think that you should just add another name on, but don't take this as a psak ask your local Orthodox Rabbi.