Royal babies are usually given multiple names – the Duke's full name is William Arthur Philip Louis – and it's likely the couple will follow suit. This will allow them to pay homage to several family members.
If the new arrival is a girl, it's thought they will in some way pay homage to William's late mother Diana, and the baby's great-grandmother the Queen. If a little boy is born, however, they could incorporate names from the Duke of Edinburgh, Prince Charles or even Prince Harry. They might even pay tribute to the Queen's father King George VI. Or they could take inspiration from Kate's side of the family.
When making their choice, the couple are likely to mindful of choosing a name befitting of a future monarch, and whatever they pick is likely to inspire other couples due to give birth in the coming few years. It is probable the pair will go for something traditional, like the names that have typically been passed down through the monarch for the past few centuries, like James, Edward, George and Mary.
They could choose something Welsh, in honour of their links to Wales – or something Scottish, to mark the time they spent at St Andrews, where they met and fell in love.
As to the question of the baby's surname, the monarchy's official website says: "For the most part, members of the royal family who are entitled to the style and dignity of HRH Prince or Princess do not need a surname, but if at any time any of them do need a surname (such as upon marriage), that surname is Mountbatten-Windsor."
George V took Windsor as the royal family's House and surname in 1917. Then in 1960, it became Mountbatten-Windsor for the direct descendants of the Queen and the Duke of Edinburgh, Mountbatten was the surname Greek-born Prince Philip took when he became naturalised in 1947.
The new arrival will of course be styled a Princess if it's a girl, and Prince if it's a boy, going on to likely become the Prince of Wales when William succeeds the throne. All male heirs usually get the title when they become second-in-line to the throne, though this is not automatic. When Charles becomes King, it is probable that William will inherit the title. The new baby will be third-in-line to the throne.
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