Need for backing up MySQL databases information_schema, performance_schema, mysql and phpmyadmin?
performance_schema: Not necessary. It may contain performance metrics that are only good for the time you were using the instrumentation. It would serve no purpose to load collected metrics from the past into another MySQL instance or into the same MySQL instance at a much later time.
mysql : BE VERY CAREFUL HERE !!! If you backup mysql, you can only reload it into the same version of MySQL it came from. In particular, the mysql.user is neither forward nor backward compatible. See my old posts:


Oct 10, 2014 : MySQL service stops after trying to grant privileges to a user
Dec 24, 2012 :Backup and restore &quot;mysql&quot; database
Apr 12, 2012 : Cannot GRANT privileges as root (I give suggestions on how to backup GRANTS as SQL text with passwords encrypted)
If you have the slow log or general log enabled to write to the tables mysql.slow_log or mysql.general_log (See my old post How to enable MySQL general log?), such a backup could be huge. The entries may also be too old to want to restore. If you wish back the mysql schema, you should mysqldump it separate from user-defined databases.

phpmyadmin : If you wish to reload a backup of phpmyadmin into the same MySQL instance it came from, that would be fine.