There may be an echo in both cross-names of the medieval legendary hero Sir Bevis of Hampton, and it may or may not be a coincidence that the site of Bewys Cross was in the tithing of Shirehampton, which was in early medieval times called simply Hampton. There is a mention in 1551 of a place called Bewehurste at Compton in the nearby parish of Almondsbury, which is of uncertain relevance.
There could be a similarity between the cross-names associated with the medieval legendary hero Sir Bevis of Hampton, and it's not clear if this coincidence extends to the site of Bewys Cross being within the tithing of Shirehampton, which was once known as Hampton. Additionally, there is mention in 1551 of a place called Bewehurste at Compton in the nearby parish of Almondsbury, whose significance is unclear.