George V tasked his private secretary, Lord Stamfordham, with finding an alternative name for the royal house. The name had to be something essentially British - but without the bloody connotations of the history of other royal names - e.g Stuart, Plantagenet etc. The process can be seen in a series of memos by Stamfordham:

The King bars Plantagenet and does not care about Tudor. Tudor-Stuart has been suggested.

These were rejected by the Prime Minister, H.H. Asquith, again stated in one of Stamfordham's memos:

'Mr Asquith has advised against Tudor, with its recollections of Henry VIII and Bloody Mary. Mr Asquith was equally averse to Stuart, one of whom was beheaded and the last driven from the throne.'

Another name was Fitzroy, but this bore too much of a resemblance of Henry Fitzroy, Henry VIII's illegitimate son. Stamfordham wrote:

He does not like Fitzroy, it hinted at wealth, but that is too foreign and is not at all liked by their Majesties who also disapprove of Fitzroy and its bastard significance.

It is said that Lord Stamfordham found inspiration (no less on the day London was raided by Gotha IV bombers) whilst working in Windsor Castle. Stamfordham, writing to Asquith to inform of his idea:

I hope we may have finally discovered a name which will appeal to you, and that is that Queen Victoria will be regarded as having founded the House of Windsor.

I've not managed to find much on why Stamfordham seemed to stumble upon Windsor as a suitable name one day working there, but being a main royal residence, it certainly had agency without the negative connotations of other regal names, and was uniquely British.