Minimize to system tray (Thunderbird)
Thunderbird cannot minimize itself to the system tray. However, you could try one of these solutions.
If you're running Microsoft Windows:
- MinimizeToTray revived Most popular. Supports Linux / Gnome
- MinimizeToTray Plus Supports Linux. (discontinued)
- Minimize On Start and Close Made to fit the Windows 7 taskbar.
- MinimizeToTray had been the most popular solution, but no longer developed and doesn't support Thunderbird 3.*. Minimize To Tray Enhancer extension that adds more features to it.
If you're using Linux:
- The FireTray extension supports "hiding to tray instead of closing". The project has been discontinued and only supports up to version 50. However, https://github.com/foudfou/FireTray/issues/242 has a link to a patched version of FireTray that works with Thunderbird 60.3. BirdTray is a possible FireTray alternative. It requires Qt5 (5.6 or higher) and is currently considered alpha software.
- If you are using GNOME, KDE, Xfce, or Fluxbox you can use kdocker or AllTray.
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:
- Fastcheck (Fastmail)
- Gmail notifier
- Notifier2 (GMail, Hotmail, Yahoo, Ymail or Rocketmail)
- Pop Peeper
- Pop 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.
- Forum thread about using Thunderbird as a pseudo shell, using MinimizeToTray Plus and FiltaQuilla add-ons.
- Minimize all windows to system tray (systray, notification area) bug report