Happisburgh is one of those Norfolk placenames that is difficult for non-locals to guess the pronunciation of, it’s something akin to ‘haze-bruh’. The Concise Oxford Dictionary Of English Placenames says about the origins of the name:
Happisburgh, Norfolk. Hapesburc in Domesday Book, Apesburga in 1150 and Hapesburg in 1272. Happing is from Haep’s people.
So, the settlement of Haep’s people, whoever Haep was. The contraction of ‘borough’ or ‘burgh’ into ‘bruh’ isn’t rare, it’s happened with Middlesborough, Farnborough and Hillsborough. I imagine it evolved quite quickly into ‘hap-is-bruh’ and then then ‘p’ was taken out over time to make it easier to say. Well, that’s my best explanation, so I’m going with that.