NOTE: I am working on a more reliable behavior for newest Android releases. In the meantime, notifications can be noticeably delayed.
Notifications for Facebook is small (0.1MB) app is for getting notified (friends, messages and notifications) from Facebook while being privacy and battery friendly. It does NOT require the Facebook Platform to be turned on. It checks periodically the mobile version of the website and creates an Android notification for any new message, friend request or general event. The notifications can be enabled or disabled per-category, with a custom sound for each.
The app and assets are free and open-source, check the app website using the link the "Developer" section at the bottom of this page.
It requires the following permissions:
INTERNET: for checking Facebook :)
VIBRATE: slight vibration for the notification
RECEIVE_BOOT_COMPLETED: so it can run after a reboot
SYSTEM_ALERT_WINDOW: used for running the update without interrupting the user