Mprs
Thanks for your interesting post. The friend from Liverpool I mentioned, and others I know, are happy to be referred to as "Scousers". Rather like Cockneys, not considered offensive at all. Scouser is the person and scouse the stew, as Chrise said. I think the origin is Scandinavian.
There seems to be several Liverpool (Scouse) accents. I think the middle class ones e.g. Lennon/ McCartney are easy to follow, but with some footballers I have heard on TV, very difficult.
I think your student would have been pleased to be called a Scouser.