Actually, for a while, BBC North covered the North east and Cumbria, North west, and the Yorkshire and Lincolnshire areas, with separate regions still in existence. The idea for creating BBC North at the time, in 1991, was to save on management and administration. In effect, it was one region, with three sub regions.
I'm not sure of the chronology but it was the way they split up England. BBC North was the 3 Northern regions, BBC South was everything from Cornwall to Kent and I think there was a BBC Midlands that included the East region. There were management for the Nations and Regions, and English regions under that in Birmingham
The regions were still full regions but the backend functions were centralised. It lasted a while too, in the mid 2000s you'd log onto a BBC desktop with a domain such as South, North, Midlands or National. There were some oddities, for example those working on the services for London were a short walk from Broadcasting House but they were being managed from Birmingham and their IT and HR departments were in Bristol!
As I say I'm not sure of what happened when but at one point there were administrative areas called 'BBC South & East' and 'BBC South & West' at one point. I think that was early 90's
The current BBC North is the name given to the BBC's operation in Salford. There is a Head of BBC North which was originally created to be in charge of the setting up of Salford.
AIUI its all changed now with the hierarchy flattened, the regions come under News although I think there is a still some sort of management in Birmingham