Thunderbird does not natively have a profile backup option, nor is it likely to have one. The arguments here can get complex, but the bottom line is that it is not just your email you should be backing up so Mozilla has left backup in the realm of Third party providers.
I am not going to get into the mine field of backup options other than to say that Microsoft has left most people high and dry with their decision to only include backup tools in Business versions of windows. That is not to say that there are not many excellent backup solutions on the market and a quick search of Google will almost deluge you in offerings.
Having said that there are some specific and free Mozilla backup options around
Perhaps the oldest of these is Mozbackup.this independently developed program is only available for windows, but it does offer some very fine grained options when backing up. Then there is the backup tool provided in the excellent Import/Export tools add-on. The author of this tool has been writing add-ons for Thunderbird for a very long time, and I would recommend a visit to the Kaosmos web site to all. There are some very useful tools there.
The import export tools offers a backup from it's options which appear in the tools menu of Thunderbird. The only complaint I would have here is that it does not offer a time of day to do the backup, but otherwise a very good start.
The developers of the Mozilla Stand alone profile manager have also included a profile backup option. This is in Beta and has been for quite some time. I have used it and found it to be quite stable and user friendly. but there a three caveats that I must mention.
- There is no installer. You must download the zip file and unzip it to a location of your choice.
- The program works with Firefox by default and the target for the shortcut you create must be modified to include ‘profilemanager thunderbird’ rather than the ‘profilemanager' that a default short cut will have.
- It does not include the Visual C++ runtime needed. This can be downloaded here.