Dion, 42, gave birth to the babies by C-section last week in Florida. The family has since been discharged from the hospital and is home.
People.com said Dion and her husband Rene Angelil named Eddy after Eddy Marnay, who produced the singer's first five records.
"He was like a father to her," Dion's representative told the Web site. "Eddy is a major influence in both Celine's and Rene's lives."
Nelson is named after Nelson Mandela, whom the couple met two years ago while Dion was performing in South Africa.
"Rene said that in just the few minutes they were able to spend with him, they were impressed by the human being he is," the publicist added. "Celine and Rene want their children to be inspired by their names because they were so inspired by these men."