Most sources seem to agree that the Red Lion is the most common pub name.
According to a sign in the Cross Keys pub, at Henley in Suffolk, the most popular pub name, prior to the reign of Henry VIII was the Crossed Keys, which is the sign of St Peter and is closely associated with the Roman Catholic church.
After Henry declared himself to be head of his newly-founded Church of England, most of the 'Crossed Keys' pubs were quickly renamed as the Red Lion, in order to proclaim their allegiance to the monarchy and to the Church of England.
Chris