I think I'll change the Wikipedia entry. Found on an obscure blog site:
The Egyptian Gods of Green Lanes
As a lifelong inhabitant of the royal borough of Haringey, I am often asked why the name of the borough is spelt as above while the neighbourhood that stretches from Manor House to Wood Green is called Harringay.
The answer is somewhat unexpected: the extra r and a are a tribute to the Egyptian sun god Ra, who has been worshipped in the Green Lanes area for at least 6000 years. His temple is now the car park at the McDonald’s DriveThru but Ra’s followers still meet on the first Monday of every month at Johnny English Potatoes and Melons under the railway bridge. To join the congregation, just go to the counter and ask for Tut.
The difference as I understand it is this: Haringey, Harringay and Hornsey are all variations of the same name, denoting the old manor, held by the Bishops of London, which covered much of the modern borough. When Tottenham, Hornsey and Wood Green were united under to form a modern London borough, a name that would denote a common historical link would be an obvious choice: but Hornsey and Harringay now meant specific parts of the new borough - so the archaic term, only previously used in recent times for a couple of streets, was more netural.