Ok, what's done up to now:
I copied the /mod/quiz folder to the /local folder
Then I renamed all occurences of the 'quiz' to 'quiz_edited' for both files' names and its content.
I failed to rename some files in /yui/build.., so I've just deleted the 'build' folder hoping that is not a big thing in core functionality...
After that, I increased the version index in /local/quiz_edited/version.php
I hoped after all that done, the site would recognize a new module to be installed, but nothing happened. I tried to purge all caches, and visited Administration/Notofocation page - no mentionings of 'quiz_edited'.
Also I went to Administration/Plugins/Local, but is said 'Nothing to display'.
If I type in address manually (like siteurl/mod/quiz/view.php?id=777 I changed to siteurl/local/quiz_edited/view.php?id=777) and go to it, the site says that I do not have permissions for '/mod/quiz_edited:..."
So could somebody give a tip please, what should I change in php files, or maybe delete something, so that the site recognizes a new module? Also, the similar question for adding capabilities...