The NZ government's ScamWatch site is warning of this exact scam:
"Scamwatch continues to receive a steady stream of reports from consumers about out-of-the-blue phone calls from scammers wanting remote access to your computer to 'get rid of viruses' or to 'fix' your computer ... The calls, which appear to be originating overseas, ask consumers for remote access to their PC to 'see if their computer is infected'. The scammer claims to be from an IT support helpdesk, or some have even claimed to be from Microsoft. If you give remote access, the scammer may go on to plant malware on the computer; or go on to offer to fix the computer for a fee – paid by credit card over the phone."The scammers are presumably using Skype or some other free VOIP service to both conceal their origins and cut their costs, hence it's known as "Vishing" - phishing by voice calls.
The scammer knew my neighbour's name and phone number - not exactly hard to find as she is listed in the phone book, but that little piece of information was nearly enough to catch her out ("How come he knew my name?" she asked me!).
Make sure your friends, family know how prevalent these social engineering attacks are. Forewarned is definitely forearmed.