How to rename the project and also its files?

I was wondering the same thing because as you said. Copy project doesn't do it. I made a new project (and folder) and imported all assignments from the project I wanted. Then closed the new project and open the original. Open Qsys and use "Save as" to save a copy to the new project/folder with the new name. That will get you the Qsys part. Then open the new project, add the qip file from Qsys and set the top design file. You may need to re-generate in Qsys, not sure. Sorry I haven't done the Eclipse file yet but my C program is rather simple because I'm learning. Hope that helps.