Land Registry are not responsible for the allocation, recording or management of house names.
As other posters have suggested Land Registry allocates a unique title number to each property as it is registered and records a Property Description to identify, in textual terms, the property being registered. A Property description will comprise of House No/Name - Road Name - Postal Town -Postcode
In England and Wales there is no legislation setting out definitive addresses, and it is not uncommon for a particular property to be described in slightly different terms by different organisations, especially where names and numbers exist for the same property. Most organisations will rely on a version of the Post Office database to match their records against a listed address.
If someone applied to Land Registry to change or record a house name against a property then we would check whether it matched that held on the aforementioned database and check any evidence supplied authorising the change e.g. a letter from the local authority authorising the change of name (or where houses are re-numbered a letter informing the owner of a change of numbering). The Post Office simply update their own database.
If you intend to change the name of your property then I would recommend contacting both your LA and the PO to enquire as to how to make the change and also whether it is acceptable - in my experience the key to any change being accepted has usually been like or similar names nearby which can cause issues for not only the LA/PO but also emergency services etc. The LA and PO though are the best places to enquire.