Interesting question. A Dangerous Method, right? I still need to see it.

The word people throw around, "psycho," is obviously a shortened, further stigmatized version of "psychotic." I don't know how long the stigmatic abbreviation has been around, but the term "psychosis" has been in use since the mid 1800s.

But psychotherapy (or psychoanalysis) didn't get its name from psychotic. They both take the Greek psyche, meaning mind or soul. Psychotherapy, therefor, is therapy of the mind/soul. I think it's perfectly fitting, and rather beautiful. Pychosis was originally shorthand for psycho-neurosis, or a mind/soul-related condition of the nerves.

All that technical stuff aside, the pure etymology of a word doesn't matter much in the light of popular understanding. I'm not sure how much stigma would be avoided with just "psychanalysis." Perhaps quite a bit? It may be that extra O that causes the unwanted association, rational or not.

Do you remember if they said, in the movie, why psychoanalysis was settled on?