From the way you ask this question, you know that the British royal family had a different name before changing it during the First World War. Before this, the British Royal family were known as the House of Saxe-Coburg-Gotha, or the House of Saxe-Coburg and Gotha, until it was changed in 1917.

The first thing to understand is that there was massive anti German sentiment in Britain, a deep hatred of the so called "Hun" at the time of the First World War. Fueled in part by propaganda, a hate against things considered German sprang up. In Leicester Saxe Coburg Street became Saxby and Mecklenburg street became Severn, but this dislike extended far beyond renaming a few streets. Germans who lived in England faced particular pressure, but even German sympathies could earn the ire of the British public, such as the War Secretary Haldane, who was dismissed from office for German sympathies (he was quoted as saying Germany was his spiritual home, which did not go over well). Anti German attacks and vandalism occurred as well. This is part of the reason why a name like Saxe-Coburg-Gotha would not have been great, and why it would have been in the monarchy's best interest to appear more British.

Another reason for this change may have occurred when German bombers began striking at London. The horror at being attacked from the air like that must have been hard to describe, and one of the most infamous incidents happened when one bomb struck a school building, killing 18 children and wounding more. Do you know what the bombers carrying those payloads were called? Gothas. Gotha bombers killed innocent children and brought terror to the streets of London.

A reason why this name change happened might have been to stay in the public's favor. Windsor Castle was linked to monarchy in Britain, it was good PR, and what European monarchs would have wanted at the time was good press. Earlier that year, in 1917, King George's cousin, Nicholas came into a spot of trouble. Tsar Nicholas was in a rather tight spot by this point, if one can believe it. This renaming may be linked to a more general attempt to try to link Britain and the monarchy, to become in some ways symbolic of Britain, its face. Edward VIII joined the Grenadier Guards (he did not, however, get to go to the front lines), Queen Mary visited hospitals and rationed in the palace, and the eventual George VI (then known as Albert) was the last British monarch to see combat at Jutland aboard the Collingwood.