What steps did you take to diagnose the problem?
If it's persisting the login information in a cookie, then your problem is either likely that
1) something is deleting the cookie, or
2) something is blocking the cookie
Did you verify that the cookie is getting written?
Find the directory where the cookies are stored. I just looked at a Windows machine that was running IE9, and it was putting them in the following folder: C:\Users\kevbert\AppData\Local\Microsoft\Windows\Temporary Internet Files. Use file explorer or equivalent to view the directory contents. Sort by date so newest is at top. Run IE9. Delete your cookies from IE9/Tools/Internet Options. Log in to BCP, making sure you click the "Log me on automatically each visit" checkbox. Go back to file explorer. There should be a new file at the top. It's a text file, you can open it in notepad, wordpad, etc. Mostly it's a bunch of gibberish numbers, but if it's the BCP cookie, it will show the text "backcountrypilot.org" in there somewhere in plain text. If the cookie file doesn't show up, then something is blocking it, and that is the problem.
Does the cookie disappear when you close IE?
Switch back to IE9. Close it. Go to the file explorer. Verify that you can still open and view the cookie text. If it is now gone, then IE is likely deleting the cookie at application exit, and that is the problem.
Does the cookie disappear before the next time you try to browse BCP.org?
Write down the filename of the cookie. The next time you experience automatic login failure, verify that the cookie is still present. If not, something deleted it, and that is the problem.