1. I just made a 0.5.6 there are not much differences from 0.5.5 but just to be sure we have the same source. it should appear as regular download in a few minutes.
Try upgrading to that one.
2, inside wp-favicon.php (the main file) you see:
//error_reporting(E_ALL & ~E_STRICT);
could you uncomment them and see if you get php error messages? Maybe you get a php error I have not foreseen while activating the plugin.
3. if you dont see any extra error messages. Try this: deactivate the plugin and reactivate the plugin
4. last option: delete the plugin via the admin panel then reinstall the plugin.
Just to be sure: you have php > 5.3 installed?