IP address keeps getting blacklisted resulting in halting of all Incoming Email. EMail host whitelists and it's good again until Cox changes IP Address?

My Email host Lunarpages keeps blocking my mom's COX IP address.  Her IP address keeps getting blacklisted resulting in halting of all her incoming Email. I call the Lunarpages, my EMail host, and they add her new IP address to their  whitelist and it's good again until Cox changes her IP Address?  This has happened more than a dozen times in the past 12 months.  All Lunarpages tech support will tell me is that the blacklisting is an automated response to 10 attempts at logging in over a 5 minute period of time. 

She uses Outlook, and she does not look at her mail every day.  Most of the time it happens when she has not looked at her email for a few days and she notes that new email has not been downloaded for a few days.  I then remote into her PC and find that she has internet access to all websites except my domain, and that is key to knowing my domain host Lunarpages has blacklisted her IP again.  No one else who uses my domain name as an email address has ever been blacklisted.

Any suggestions on how to fix this?