People looking to make money from domain name auctions were quick to register leoblair.com and babyleo.co.uk as soon as Downing Street revealed the child's name.
A primary school teacher from Norfolk, Diana George, said she registered leoblair.com and leoblair.co.uk after the publicity surrounding prices paid for popular domain names.
If the prime minister and his wife want to buy back their son's identity in cyberspace, they could face a hefty bill.
Teacher's net
"When I heard the name on the news, I registered it straight away," Diana George, 47, of Hickling, Norwich told BBC News Online.
"It occurred to me that there would be a lot of interest in it and it might be of use to someone."
Mrs George has not yet decided whether to approach the Blairs.
"The thought had crossed my mind," she said. "But if I do sell it I will give half to charity."
Mrs George, who describes herself as a cyber-novice, started dabbling in cyber-squatting after reading about the colossal sums paid for domain names, which can be registered for about £30.
Cybersquatters, people who occupy a web address that might rightly belong to someone else, make the most of big companies wanting snappy names for their sites.
Payouts have included $275,000 for RomanCatholics.com, $500,000 for Drug.com.
Mrs George has also registered vodafoneallcalls.com as part of her cyber-investment.
Coup for squatters
Slower off the mark on Saturday was Glasgow IT consultant Charles Sweeney.
He was left with the option of babyleo.co.uk. and babyleo.com.
"Someone had obviously had the same idea," he said.
Mr Sweeney, 37, of East Kilbride, said he had been inspired by news of the coup in Fiji led by George Speight.
"I thought that was an unusual name and registered georgespeight.com," he said.
"I did babyleo.com at the same time."
Mr Sweeney registers popular names to entice web surfers to his own sites for renting houses and flats.
"The next pop star or footballer who has a kid will rush to get the name," he said.
The news should come as no surprise to Tony Blair, whose own name has already been registered on the internet.
Even rumours of a pregnancy can prompt a rush by cyber-prospectors.
Babyblair.co.uk and babyblair.com were registered within days of the news that Cherie was pregnant in November.
But a Downing Street spokesman said on Sunday the couple had more important things to think about.