Birdtray provides systray notifications for Thunderbird. It displays the count of unread mail, hides the Thunderbird window when not in use, and restores it on clicking the tray icon. It also provides a context menu with commands such as starting composing a new mail.

It is a nasty hack -- an external process looking at Thunderbird's insides, it suffers from problems like noticing new mails only after a delay, having to restart Thunderbird just to hide its window, etc -- you'd want to use an extension like firetray instead -- but, it is likely that support for Thunderbird XUL extensions will be dropped soon, possibly by the time you read these words.