The birth had been long and difficult. Denise Olivio requested a cesarean section, she said, but was rebuffed until a doctor decided it was an emergency.

The baby, GianRaul, came out purple, and it took doctors 15 minutes to locate his heartbeat. Hours later, Ms. Olivio and her husband — exhausted but relieved that their son had survived — said they saw their doctor and midwife being pulled aside by a man in a suit who they believed was the hospital’s lawyer.

“We locked eyes,” Ms. Olivio recalled, “like, ‘Oh, I think we need to talk to someone in a suit, too.’”

The family sued the hospital over the brain damage that had left GianRaul unable to walk, talk or breathe on his own, and it took seven years before a $6.5 million settlement was reached. In that time, he was hospitalized repeatedly for seizures, respiratory and digestive problems, even at one point receiving hospice care. Each time, GianRaul pulled through, giving rise to his nickname: G.M.O., a partial acronym for God’s miracle on earth.

Under a quirk of New York law, the family received just half of the $6.5 million settlement — used to pay the parents’ lawyer, unpaid doctors’ bills and Medicaid, as well as to set up a trust for GianRaul’s care.