If you keep your desktop on 24/7 as many people do, then you might keep your email client (thunderbird) open in the background, checking for mail every 5 minutes so you know when you get new email. Since your storage quota is only 250mb, you delete email from the server automatically on download. Now suppose you want to take an email with you to a community meeting. You can print it out, or you can email it to yourself at a different address. Or if you were able to link up your phone with your computer's storage, you can just search up the email on demand, find related messages, and make changes to drafts for your community mailing list.
If you could connect to thunderbird with an IMAP client, then this would be possible.
However I see two ways to let my phone read old mail
1. Set up my own IMAP server at home and deliver email to it, then program some filter system to mimic what I'm doing in Thunderbird.
2. Access Thunderbird directory on my laptop over IMAP. That IMAP server would be on my laptop or (better) on some always on server at home that uses the same file format as Thunderbird. I would rsync it with my laptop.
I admit that number 2 is weird. Number 1 is the proper way but I have so little incentive to do it that I'll probably stick with my current setup forever.