A spokesperson for the music mogul, who became a father for the first time on Valentine's Day, confirmed that the youngster's name is Eric Philip.
Both names chosen by Simon and his socialite girlfriend Lauren Silverman are a tribute to the X Factor judge's late father. Eric Selig Phillip Cowell died of a heart attack in 1999 at the age of 81.
Fans have also said that the name Philip is in recognition of Simon and Lauren's close friend, Topshop magnate Sir Philip Green.
Sir Philip is said to have helped the couple while Lauren was divorcing her husband Andrew Silverman.
"Simon wanted to pay proper tribute to the influence of his late father in his life," a source told The Mirror. "Eric Philip is clearly a very proper tribute to the man.
"But there's no doubt he also wanted to pay some kind of tribute to another Philip who has been a major influence in his life, Sir Philip Green," added the source. "It's tremendous he's been able to pay respect to two different role models with his baby son's birth."
Simon and Lauren's son was born at 17:45 on 14 February, with the proud new dad declaring on Twitter that his baby was "healthy and handsome."
Lauren, who is already mother to eight-year-old Adam from her marriage to Andrew Silverman, was said to have had her mother Carole Saland and sister Nicole Davis by her side for the birth.
On Saturday Carole, Nicole, Simon and Lauren's dad Steven Davis were seen arriving at Lennox Hill Hospital to be with the new mum.
The 54-year-old music producer beamed and waved at the waiting reporters as he walked into the hospital carrying a bag reportedly full of gifts for Lauren.
Simon's mother Julie Brett is said to be eager to meet her grandson back in England.