Frank Steinhauer solved his problem and asked me to post this to the list:
* The problem is the order of \usepackage. apacite must not be included before babel - but LyX does it. So I changed the generated .tex file, and did the latex and bibtex run by myself. Maybe Lyx could be changed to support another \usepackage order.
* Oh, apacite needs a BibTeX database - another thing I noticed during the last week...
-- Paul (for Frank)