The most popular choices so far are George, James and Alexander, while Arthur, Richard and Charles are also possibilities, according to Ladbrokes.
HELLO! Online has asked its readers on Facebook and Twitter to play the name game, and give their predictions on the royal baby's full name.
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@RoyalGawker: "George Francis Philip Arthur: Francis for Michael Middleton and Diana, and Philip for Duke of Edinburgh."
@Melissa24K: "'Michael Philip Spencer Charles': a nod to the grandfathers, great-grandfather and Will's mum."
@NeinNeinOKFine: "I love the name Henry & it'd be a nice gesture towards new Uncle Harry ;) Time for another King Henry, a great one." v @RealCSI: "Spencer would be a great middle name. It would pay homage to Diana and carry on the legacy of Diana who would be over the moon!"
@LaurenDrake2: "George is what I'm thinking — a traditional English name".
Libby Barebum Fairbairn: "I think it will be Alexander. It's a more modern name now and they are the most modern royals at present. They are fun and funky people and I think that will be reflected in the baby's name."
Glen Jevon: "I reckon they'll go for Phillip. Not only because of his great granddad, but because he was born on the same day as distant relative Prince Phillip the Handsome, who was King of Castile in Spain."
Katherine Stocker: "George, William, Charles, Michael or Francis. George- traditional, William- after his dad, Charles- after Prince Charles, Michael- after Kate's dad and Francis- after Princess Diana Francis. But maybe a Welsh David or Andrew, as the couple met at St. Andrews."
Christina Iwanitza: "It has to be a posh name. Anything out of the norm will just not be allowed. I guess the baby could be named after William's father or Kate's, so maybe it will be called Charles or Michael. But in the end, only they will decide… it's their son."
Chez Gordon: "I would love the new baby to be called Louis — a very traditional name. So old that it's new again!"