Minimize to system tray (Thunderbird)
From MozillaZine Knowledge Base
Thunderbird cannot minimize itself to the system tray. However, you could try one of these solutions.
If you're running Microsoft Windows:
If you're using Linux:
If you get two windows and can't get rid of the second one try disabling the add-on, exiting Thunderbird, starting Thunderbird, closing one of the windows, re-enabling the add-on, exiting and restarting. 
With the following utilities you can minimize not only Thunderbird, but any program to the system tray.
Email notification utilities
Some email notification utilities for Microsoft Windows that support the system tray:
New Mail Attention adds support for a message filter action to call attention to the main window when a new message arrives. It supports several operating systems and the MinimizeToTray Plus add-on.