After receiving loads of these calls, I called the other number listed above (0870 850 1281) and got through straight away.
I was asked for my credit card number, and explained that I don't bank with Halifax, but keep getting called by their automated system, asking for someone who I've never heard of.
After confirming the number and the name that was asked for, the operative did something on his computer and told me that he had deleted the number, and that I should stop getting called. He did explain that I may continue to get calls for another 24 hours as their systems take a day to update, but after that it should stop.
The reason suggested that I was getting the calls was that the number had been registered previously by one of their customers, who had probably not updated their details with Halifax.