Suggesting that although the Duke and Duchess of Sussex are seen as thoroughly modern royals, the public does not believe the new parents will break the mould when naming their child.
Members of the royal family traditionally choose regal favourites for their youngsters.
The Duke and Duchess of Cambridge toed the line with George and Charlotte, and Louis has key family connections for the Windsors.
But Meghan is an American ex-actress who may want to draw on her own personal inspirations when settling on a name for her baby.
Harry and Meghan’s youngster is not a future monarch nor an HRH and will only be seventh in line to the throne.
Canadian-born Autumn Phillips and the Queen’s grandson Peter Phillips chose a non-traditional name for their first child Savannah – the monarch’s first great-grandchild – in 2010.
When she ran her lifestyle blog The Tig, then Meghan Marke wrote that one of her favourite baby names was Grey, which works for both a boy or a girl.
For the Sussexes, Coral has Alexander leading the odds, while a rush of bets has been placed on Spencer, in honour of Harry’s mother Diana, Princess of Wales, whose maiden name was Spencer.
John Hill of Coral said: “Following a flurry of bets last evening, Alexander is now the new favourite at the top of our name betting and the one we are fearing the most.”
Alexander is 4/1 at Coral, while Arthur is 6/1, and Albert, James and Spencer are all 8/1.
“The interesting name which we have taken plenty of money on is Spencer as we did not lay a single bet on this name before the Duchess of Sussex gave birth,” Mr Hill added.
Harry said he and Meghan are still deciding on a name.
As he announced the birth at Windsor, the Duke added: “Still thinking about names.
"The baby is a little bit overdue, so we’ve had a little bit of time to think about it.”
At Paddy Power, Alexander and James are joint favourites at 7/2, followed by Arthur at 6/1, and Oliver at 12/1, which has fallen swiftly from 33/1 following the baby’s birth.
William Hill has Arthur at 4/1, James and Alexander at 5/1, Albert at 10/1 and Philip at 11/1.
Spokesman Rupert Adams said: “We have had the first significant move since the announcement and Alexander has been halved in price from 10/1 to 5/1.”
Coral’s outside odds include Apple at 200/1, Boris at 250/1 and Donald at 1000/1.
The baby, who is seventh in line, is 2-1 to become an actor, and 500-1 to end up on the throne.