Thunderbird uses the IMAP protocol to download messages. There are command-line IMAP utilities with which you can automate connecting and downloading messages from your account. Look at getmail (http://pyropus.ca/software/getmail/) as a possible option. You can very easily automate this with a cron job (How do I set up a Cron job?).
As an example you could create a ~/.getmail/getmailrc file with something like:
type = SimplePOP3Retriever
server = pop.example.net
username = your.username
password = mailpassword
type = Maildir
path = ~your-username/Maildir/
Then in /etc/cron.d add a file "retrieve-email" with something like this:
15 10 * * * your-username getmail
his will cause getmail to download your email every day at 10:15 AM and put it in the Maildir directory.
PLEASE don't use these examples as they are; they are incomplete and illustrative only, I haven't tested them. Read the documentation I referred to so you can understand which getmail parameters you need.