I’ve now discovered a very useful website: Nottingham University’s Key to English Place-names. This is much more cautious about the derivation of Countisbury than AD Mills who simply quoted arx Cynuit c. 894 (that’s Asser).
The problem is that when the historians were looking for somewhere on the Devonshire coast that might have been the site of the ‘Battle of Cynuit’, they fixed on Countisbury as the possible place, due to its situation and the similarity of the name; the place-name specialists then took the modern name as deriving from Cynuit, because that was ‘where the battle took place’. That was the wrong way round: if the specialists had first derived Countisbury from Cynuit on existing documentary evidence, then the historians could have come along and said, ‘Well, that solves the mystery of where the battle took place, then.’
Anyway, the Nottingham site says of Countisbury’s derivation: “Uncertain. Perhaps, ‘fortification called/at *Cunet’, a Celtic place-name of unknown meaning. Alternatively, it has been suggested that the first element is a Celtic personal name, ‘Cynuit’.”
But where did this ‘Celtic place name’ *Cunet’/cunẹ̄ti̥ū come from? It resembles the personal name Cunetus, found in the medieval document referring to Cuneti confessoris as the patron saint of Llangynwyd in Glamorgan, where the original form would appear to be the British/Welsh/Celtic Cynwyd or Cynuit; the later Cunetus is probably a late/neo-Latin back formation; but this is the personal name, not a place name.
It looks like the Roman place name Cunetio, which according the Antonine Itinerary (3rd c. AD) was some 35 Roman miles from Bath and has been uncovered by Mildenhall in Wiltshire. The origin was the British/Celtic name of the river on which the Romans built their town – now the River Kennet. This river name may well be represented as having been something like *Cunet’/cunẹ̄ti̥ū in Roman times. But Cunetio is nowhere near Countisbury.
There are several river names with the Celtic element Kenn or Kennet, but they aren’t near Countisbury either. It does seem as if this *Cunet’/cunẹ̄ti̥ū would, phonologically, give Cynuit – but there is still nothing to link it to Countisbury.
As I see it, the Nottingham site is more circumspect regarding the derivation – which leaves us … where we were.