Paddy Power has Diana at 8/1, with many believing Harry and Meghan will pick this for a girl in tribute to the duke’s late mother Diana, Princess of Wales.
William Hill has Alexander as its favourite at 7/1, while Ladbrokes said Victoria – the name of the nation’s second-longest reigning monarch – is its favourite at 8/1.
Rupert Adams, spokesman for William Hill, said: “We took eight bets in the first minute today, and given the popularity of the couple this could be bigger than the original market we had for Prince George.”
A Paddy Power spokesman said Diana was the firm’s clear favourite.
The princess was killed in a car crash when Harry was just 12 years old.
“Diana is an obvious favourite with our punters so far, and it wouldn’t be the first tribute the couple have paid to the late princess, while Arthur and Alice have been popular picks too,” a spokesman said.
Odds that the newborn will have red hair just like Harry are 3/1 at Paddy Power, who also have odds of 250/1 that the baby will share US president Donald Trump’s first name.
Meanwhile, Ladbrokes are offering 100/1 that the baby will be named after Meghan’s outspoken half-sister Samantha Markle.
Jessica Bridge of Ladbrokes said: “Given this child will be in line to the throne, it’s probably a safe assumption that the couple will opt for a regal name that has been used in the royal family over the generations.
“However, with Meghan’s USA roots it would be remiss of bookmakers to rule out an American moniker featuring somewhere.”