The royal names are based on a set of strict rules which are being followed for ages now. But Diana was very particular that she would pick both her sons' names.
Picking a name for a royal baby is ceremonious and a very different process as compared to us civilians, especially when it is about the British royal family. They have long abided to a set of strict rules to zero down on a moniker that would be in sync with the monarchical heritage. But royals too have a lighter side to them and when it comes to Princess Diana, we are bound to get a heavy dose of extraordinarily ordinary charm.
The people's princess wanted to ensure a normal upbringing for both her sons and that reflected in many of her actions. For example, escorting them to their school herself, keeping aside a mother-son playtime where she found back her own childhood again, trying to protect them from the paparazzi, etc. Therefore, it is only plausible that she would want them to have common names rather than the throne-worthy ones.
In a 1992 interview with her biographer Andrew Morton, she revealed that it was she who had picked the names of both her sons. She was asked "who chose [Harry's] name?", to which she replied: "I did". Had it been the Prince of Wales taking the charge of their names things would have been very different.
"I chose William and Harry, but Charles did the rest. He wanted Albert and Arthur, and I said no. Too old!" she added. Now that we think about it, William and Harry go perfectly well with their down-to-earth and humble personalities. However, the Prince of Wales did have a say in their full names. William's full name is William Arthur Philip Louis and Harry was named Henry Charles Albert David, on birth.
According to Her, this also means that Prince William, if and when ascends the throne can opt for any of these names as the King of England.
Interestingly, Prince Harry's little prince has also been given a not-so-royal name. This is another way Prince Harry exudes the same qualities of his mother. While the little prince's christening was a heavily guarded affair with only close friends and family members attending it, his first name reveals the same thought the Princess of Wales had put for the names of her children.
Archie Harrison Mountbatten-Windsor is the first-born child of the Duke and Duchess of Sussex who was christened five days after Princess Diana's birth anniversary. This proves that the Duke of Sussex not only misses her mom but tries to follow in her footsteps whenever he can.