Your own sessions never light it, and it relies only on recent data from the SyncNow server. Pointing at it says Someone is working on this project. In the sidebar, a green dot on the Updates icon means a new version of SyncNow is available.
The project's dot goes away by itself when nobody else is working on the project any more. To stop it for one project, open Project settings (the Settings tab, or Project settings in the ... menu next to the project's name) and, under Notifications, untick Show notifications: the project also stops notifying its new posts and stops counting in the number on SyncNow's taskbar button and on its icon in the notification area, while the grey number of unread updates next to its name in Projects stays.
Show notifications applies to everyone who manages the project: it is one choice for the project, kept on the SyncNow server, so unticking it silences the project for the owner and for every manager, on every PC; the change itself is not posted in Activity and notifies nobody. The dot on Updates cannot be turned off: it stays until the new version is installed.