And according to the book, the Queen - in line with Royal Family traditions, and the advice of former prime minister Winston Churchill, her grandmother Queen Mary as well as her own mum, the Queen Mother - refused to give their children the Mountbatten name.
Instead their surname would remain as Windsor.
However the resulting argument between Prince Philip and his wife reportedly left the Queen in tears, reports The Express.
By 1952 when her father King George VI died, and Princess Elizabeth (as she was at that time known) became Queen Elizabeth II, she had two children.
Prince Charles, who was born in 1948 and Princess Anne, who was born in 1950.
As she came to be Queen, the monarch issued a public declaration and confirmed that “her children will be styled and known as the house and family of Windsor”.
Something which was said to frustrate Prince Philip.
According to the book, he felt “like a bloody amoeba" as he was the “only man in the country not allowed to give his name to his own children”.
By the time the Queen was pregnant with Prince Andrew in 1960, she had a change of heart and she told then Prime Minister Harold Macmillan that "she absolutely needed to revisit" the issue of the family name because "it had been irritating her husband since 1952".
In the biography, an entry in Mr Macmillan’s diary is cited, where he wrote: “The Queen only wishes to do something to please her husband – with whom she is desperately in love.
"What upsets me is the Prince’s almost brutal attitude to the Queen over all this.
“I shall never forget what she said to me that Sunday night at Sandringham.” reports The Express.
On February 8, 1960, 11 days before the birth of Prince Andrew, they reached a compromise.
The Queen made a new declaration in Privy Council saying that she had adopted Mountbatten-Windsor as the name for all her descendants who do not enjoy the title of His or Her Royal Highness.
Meghan Markle and Prince Harry's baby is one of the British royals who uses this name. The new bub was born on May 6, and named Archie Harrison Mountbatten-Windsor.