We've had this issue reported to us by a client and since I don't have a webdav repository set up I haven't been able to confirm or test the issue. However I think I've isolated the problem and my proposed patch seemed to fix it for them so here it is!

When viewing the webdav repository in the filepicker they get the error:

ERROR: Invalid JSON string 

Warning: strpos(): Empty delimiter in /var/www/htdocs/repository/webdav/lib.php on line 146 

{"dynload":true,"nosearch":true,"nologin":true,"path":[{"name":"WebDAV","path":0}],"list":[{"title":"test","thumbnail"...

From what I can tell, if the path is just a single slash '/' path will end up being an empty string as the line before is:

$path = rtrim($path,'/');

This doesn't seem to affect 2.3 following the changes made in http://tracker.moodle.org/browse/MDL-32497.

Sorry I can't provide any more information!