This is based on the post at http://www.freepbx.org/forum/freepbx/users/email-on-missed-call-request, which references an article at http://www.theschmandts.org/blog/2007/05/05/email-notifications-for-missed-calls-in-asterisk/ which describes a technique that as written would not work in FreePBX. However, my thought is that you could add a yes/no option in the extension settings, under "Voicemail & Directory", called "Email on Hangup" which if checked would send an email if a caller either hangs up before the voicemail greeting is completed, or leaves a message that is too short. That section would be the logical place to put that feature because you already have the delivery e-mail address, and presumably the ability to send e-mail.
The email should contain the extension called (and perhaps the original DID called if that information is still available), the Caller ID name, Caller ID number, and the date and time of the call (although the latter could possibly be deduced from the e-mail date/time).
