Where there any other names considered? Where did Windsor come from?
'Windsor' came from the iconically English town of Windsor, built around the eponymous Windsor Castle, which was originally constructed in the 11th century, and is today the longest-occupied royal residence in Europe, having housed English monarchs since Henry I. The choice was motivated by the centuries-long association of Windsor with the English monarchy.
As to exactly what happened: George V charged his Private Secretary, Arthur, Lord Stamfordham, with identifying a suitably British and genteel replacement for Saxe-Coburg-Gotha. Lord Stamfordham initially evaluated the names of prior English and British dynasties, such as Plantagenet, Tudor, and the hybrid Tudor-Stuart, as well as names like Fitzroy, but each was vetoed by either the prime minister (Herbert Asquith), or else George V himself.
After a number of rejections, and the bombing of the UK by aircraft from Gotha, Lord Stamfordham decided on the name Windsor while in the castle itself, and made the recommendation, which was rapidly accepted.