The names of pubs, roads and even hospitals are testament to the long association of Archway with Dick Whittington (c. 1354–1423), the man who was famously three times Lord Mayor of London.
There is a very thorough piece on the subject of his memorial on the fascinating IanVisits site The Mystery of Dick Whittington's Stone (even if the stone is described as being in Highgate).
Certainly by the early 19th century there was a clear belief that Whittington walked back home via Highgate Hill. That was why when his original almshouses in the city were demolished, the replacements were built as close as possible to Highgate Hill. And while some have claimed it unlikely that he would have walked north to reach his family home in Gloucester, because that is west of London, the explanation is that it was a time when people still used the old Roman roads and there was no direct route.
The current A40 is made up of coaching roads which only appeared in the 1700's. Whittington lived at least two centuries before that so from Gloucester to London would have taken Akeman Street go St Albans, and then the high ground road (avoiding mud in bad weather) to Barnet and Highgate and on to the City of London. When Whittington was thinking of returning home he would have retraced his steps, walking up the hill from which he could look back down on the City and hear its church bells.
Whittington was the younger son of a rich family but by the 1600s stories were turning him into a scullion or poor servant boy who miraculously made good. No-one knows if he really had a cat and the one attached to the Victorian milestone on Highgate Hill was only added in the 1960s.