I checked this on Wikipedia.
I first went to the article on the "House of Hanover", on March 28th, 2024. It claims that "the formal name of the house was the
House of Brunswick-Lüneburg, Hanover line" and notes that "originating as a
cadet branch of the
House of Welf in 1635, also known then as the House of
Brunswick-Lüneburg". So the name of the dynasty in Great Britain was the name of the town where they had their palace.
Then I tried "House of Windsor", same date, and only got that the House was named after the royal residence. The house of Hanover was also named after the residence. Likewise, "house of Saxe-Coburg and Gotha" is really "House of Wettin, Saxe-Coburg and Gotha branch". The current royal house of Great Britain would likewise be the "House of Oldenburg, Windsor branch."
Really all George V did was to emphasize the main residence of the house, which is Windsor, and not the Germanic name for the entire family, which was Wettin.
The most plausible alternative name would have been "House of Westminster". The main urban royal residence is St. James Palace, not Buckingham Palace.