With previous favourite options such as Diana and Allegra no longer in the running, bookmakers have seen predictions for names such as Philip and Oliver surge in popularity.
Currently, Alexander and Spencer are the most popular choices with 4-1 odds, according to Coral, followed by Arthur at 6-1.
Arthur has been used several times in the royal family tree, including as the middle name of the Duke and Duchess of Cambridge's son Prince Louis.
Other less-likely possibilities include Donald, with 200-1 odds, and Boris at 250-1.
The Duke and Duchess announced the arrival of their first child on Instagram, where they revealed that both the duchess and the newborn are "healthy and well".
“We are pleased to announce that Their Royal Highnesses The Duke and Duchess of Sussex welcomed their firstborn child in the early morning on May 6, 2019," the statement reads.
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"Their Royal Highnesses’ son weighs 7lbs 3oz."
For all the latest news on the royal baby, follow The Independent's live blog here.
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