I am thinking I have misunderstood your question as my answer seems too simple, so please forgive me if this is the case. You should be able to rename either a document or a folder by clicking on the name with your cursor, the name is then highlighted and you can type your new name over it.
well, I've done it niceguy2's way ... tried what you said bartonian (feeling very foolish as I did so ..) but nothing doing ...
on my old mac (snow leopard) I got the option to 'save as' in preview and could then rename at that point but with lion all i'm getting is 'save a version' and then it goes into downloads and then i have to shift it over to the right documents folder
Click once to select the file, it should then be highlighted. Then click again on the file name underneath the icon, you should be able to edit it.
Using Preview "Save a version" does not copy the image, it backs up a copy of the image as it is at that point in time. If you click at the top of the Preview Window on the file name you should get a little drop down appear with a "browse all versions" option. If you click that you can see every version of the file going back to when Lion was installed. The system automatically saves versions periodically, "Save a version" just forces the process.
If you want to copy the image then select "Duplicate" from the File menu. That will open up a new Preview Window and then clicking on the File menu will now show a "Save" option as it's never been saved before. That is the only time, I think, that you'll get a "Save" option.\
Or to copy an image you can just use the "Duplicate" option when right clicking on a file in Finder.