Most of the users have "read" rights to the Portal. Some users need
to be able to upload documents to the library and then modify
a "content viewer web part" and add a link to the document. I know
that "author rights" are needed to upload docs, but what about
modifying a web part?

Do users need to have coordinator rights to do this, or is there a
way of configuring the web part to allow users without coordinator
rights to modify them?

I don't particularly want to give a whole bunch of users coordinator
rights, just so that they can make changes to a web part...

I have also tried giving them author rights, but they are still
prompted with a username and password, when they attempt to save the
changes to the web part.