The child came unexpectedly early into the world on Tuesday while the couple were on holiday in Cornwall.
Endellion refers to the village of St Endellion, close to where they are staying and which Mr Cameron has visited over the years.
Last night Downing Street said that the couple had received congratulatory messages from the Queen, the Prince of Wales, Gordon and Sarah Brown and French President Nicolas Sarkozy.
Mr Cameron spent his daughter's first night at the Royal Cornwall Hospital in Truro, where she was delivered by Caesarean section.
The couple's two elder children, six-year-old Nancy and four-year-old Arthur Elwen, yesterday visited their sister for the first time.
The director of nursing, midwifery and allied health professions at the hospital, Christine Rashleigh, said she hoped Mrs Cameron would get to spend as much time as possible with the new baby before returning to the "cut and thrust of working and supporting the Prime Minister".
She said: "July and August are really busy times for us. We double our population in Cornwall during the summer due to the holidaymakers and we are always ready for that eventuality.
"The nurses will be making sure that Mrs Cameron rests as much as she possibly can and pays attention to her other children as well as her new baby because that's going to be really important for the family to bond as a whole, and certainly Mr Cameron will play a very key part in that."
Mr Cameron had said he was keen to include a reference to the county of her birth in his daughter's name and the choice was met with delight in St Endellion.
Prebendary John May, who lives at the village rectory, said it was "brilliant news" which had quickly spread through north Cornwall.
"It's absolutely fantastic," said Prebendary May who heard while at a church fete three miles away at St Minver.
He said he was not sure if the Camerons had visited St Endellion but described the village as a small but special place – little more than a church on a hill with a rectory behind it and a couple of farms and cottages.
But he added: "People regard it as just a remarkable place.
"Every church is remarkable but some places have some kind of magic about them, they seem like a sacred place or holy ground."
The former Poet Laureate Sir John Betjeman famously described the village church as looking like a hare, writing: "The ears are the pinnacles of the tower, and the rest of the hare, the church, crouches among wind-slashed firs."
Experts predicted the naming of Florence was likely to herald a flurry of children born with the same name.
According to the most recent statistics, Florence is the 94th most popular girl's name and was given to 608 babies born in 2008.
But it has recently soared in popularity after pop sensation Florence and the Machine stormed the charts and charmed the nation.
The Camerons suffered tragedy last year when their elder son Ivan, who suffered from cerebral palsy and severe epilepsy, died aged six.
The new baby is only the third born to a serving prime minister since 1849.