While it's possible the reader's mother may have failed to cancel the account before the free trial expired, what really has the reader upset is Earthlink's unwillingness to honor any of their attempts to cancel the account.Here is the first problem. Failing to cancel the account on time. This automatically makes it difficult for you to claim to be blameless in this matter. You should have signed her up with a billed ISP from the start, preferrably a smaller local one that gives a care about its customers.
When a situation like this occurs, the first thing you should do is inform the customer service representative you call that you are recording the call; state your name, the date, and the time. Ask them for their name and their employee ID. Inform them of your intent to cancel; if they attempt to deflect it, tell them firmly "I am informing you that I wish to cancel NOW. You are legally obligated to honor my request." Get a confirmation number (or cancellation number, or whatever this ISP calls it), and document all of this. If they fail, you have a paper trail. Nothing replaces good documentation; if you don't document it, it didn't happen.
One other method is to send a written request via registered mail (requires a signature) with all of your pertinent account details. In your case, it has gotten far enough that I would pay $50 and have an attorney write up a short letter on your behalf indicating what has happened, your insistence that the account be canceled, and that you be only obligated for the amount you truly owe.
Should you have to do a lot of this garbage? No, but it has dragged on long enough that this is the only way. And in the future, remember that if you have a free trial, you MUST cancel within that time period or you're on the hook Getting off the hook once you're on is what has made this far more of a pain than it ought to be.