As an immediate solution I suggest switching to an online email system such as GMail. This can be done easily by redirecting the account or using gmail fetcher to check the external server, outgoing emails can be sent with the existing from address so there is visible change.
How does BigPond work? I assumed it would be an web based online system, maybe that is a bad assumption my part.
In which case, if it is something like outlook on his computer talking SMTP/POP/IMAP need to make sure it is secure, like other people have mentioned.
It is is a desktop computer, would be surprised if it is using WIFI, but something I should check.
Are compromised routers common?
There a number of possibilities here. I don't like to suggest random changes unless I can get some proof that the change addresses the actual attack vector. e.g. changing to a gmail account would not help if there is some sort of hidden keyboard logger on the computer. Apparently it is a bluetooth keyboad, how secure are bluetooth keyboards? Or if my relative changed an obvious password to another easy to guess password.