Since 1865, the official name of the village has been BURNMOOR. It officially came into being when a new parish was formed by an order issued by the Queen in Council (ie. the British Monarch acting with the advice and consent of the Privy Council) on 28th December, 1865. The official title of the document was:
“For constituting the District of BURNMOOR out of the Parish of Penshaw, sometime part of the Parish of Houghton-le-Spring, in the County and Diocese of Durham”
The full document can be found here: https://sites.google.com/site/allthingsbournmoor/birth-of-bournmoor---legal
The same document also makes reference to BOURN MOOR which suggests that the two can be used interchangeably:
"The District of Burnmoor, being:
‘All that portion of the parish of Penshaw, sometime part of the parish of Houghton-le-Spring, in the county of Durham and in the diocese of Durham, which is comprised within and is coextensive with the limits of the contiguous townships of South Biddick and of Burnmoor, otherwise called Bourn Moor.’"
The very oldest maps of the district make reference to “Burn Moors” (1768 map by Armstrong), “Burn Moor” (1788 map by Gibson) and “Bourn Moor” (1843 map by Bell). See: https://sites.google.com/site/allthingsbournmoor/burnmoor---old-maps
It is also interesting to see how BURN and BOURN are used almost interchangeably in the early trade directories for the area. See: https://sites.google.com/site/allthingsbournmoor/j-bournmoor---in-directories
The two main theories regarding the origin of the name are:
BURNMOOR / BURN MOOR: The moorland through which the Moor(s) Burn runs.
BOURNMOOR / BOURN MOOR: “BOURN” is an old English word meaning a limit, edge or boundary, so the moor at the edge/boundary, possibly of the Lambton Estate.
Both of the above support the fact that this was once an area of moorland.