When one traces their family tree, rules usually stay the same in relation to a person's surname. This title is passed on through the father's line. If you are a boy, it will remain unchanged for your whole life. If you are a girl, it will change if you marry and choose to take on your husband's surname. While times are changing and women have more freedom to keep their surname after marriage or to hyphenate it with their husband's surname, for the royal family, things are entirely different.
Take the princes, William and Harry, for example. They are both brothers so they should hold the surname of their father. Therefore, William and Harry's surname is taken from the title of their father, Prince Charles of Wales, according to Business Insider. But, as the royal family's official website points out, in the British royal family, the surname is determined by one's title rather than the surname of one's father. As a result of this, royal family members can quite often have their surname changed over the course of their life.
And this has certainly happened for Prince William and, more recently, Prince Harry. When William married Kate Middleton, he was given his own title, independent of the one his father carries. As a result of this, Prince William is now known as the Duke of Cambridge. Harry, on the other hand, with his marriage to Meghan Markle, was appointed the title of Duke of Sussex.
Now, even though they are brothers and both grew up with the surname of Wales, they are now titled with different surnames. Which means that their children will also be gifted with these surnames rather than that of their father's birth surnames, as Hello! Magazine points out.
So, William's children will bear the surname of Cambridge and any children Harry and Meghan have will have the surname of Sussex. Of course, when these children grow and marry, these surnames may change just as their fathers did when they married.
While a royal wedding quite often signifies a name change if the couple is gifted a title such as duke or earl, this is not always the case. There are only so many titles available, and when they are used up, royal family members have to wait in line for a title to become available -- which usually means the death of a relative, or someone relinquishing the title and passing it on.
In addition, if you are a royal family member who is lucky enough to be titled Her or His Royal Highness, there is no need for a surname to be used at all. However, as the royal website points out, those bearing that title can use the surname of Mountbatten-Windsor if they so chose.
This means that although Prince William's children are currently carrying the surname of Cambridge, they also have the option to remain without a surname or to use Mountbatten-Windsor instead.