Henry Dunham was the son of Peter Bishop and Fauxlivia, who became pregnant during her time with Peter in the other universe. Her mother, Marilyn Dunham, was worried that he had an in-utero disease, Viral Propogated Eclampsia (VPE), that would result in the death of either the baby or his mother during birth. To prevent this occurrence, his mother was abducted by a group of doctors that sped up the pregnancy under Walternate's orders. She was, after some time, able to escape the place where she was being held and gave birth to him, with the help of Henry Arliss Higgins and Lincoln Lee in a Chinatown store. Because the pregnancy was sped up, the disease was not fully developed, allowing for both Henry and Fauxlivia to survive the birth. Under Walternate's command, the doctors sampled his DNA and delivered it to Walternate. ("Bloodline")
He was later seen in a baby stroller being taken for a walk in a park by his mother and nanny. When his mother had to go, Mary took over and told Fauxlivia that he would be safe with her. Before she leaves, Fauxlivia says goodbye to Henry - the first time his name is used. Presumably, she named her son Henry in honor of the help Henry Arliss Higgins gave her in delivering him. When Fauxlivia planned to betray Walternate, she told Lincoln to deliver Henry to his grandmother Marilyn for care if she did not return by the next day. ("6:02 AM EST")
Because Henry's father, Peter Bishop, ceased to exist following the events of The Day We Died, Henry Dunham's status in the original timeline remained unknown. In the new timeline following Peter's erasure, Henry no longer exists, as Peter had died as a child.
When Peter went inside September's mind, the Observer informed Peter of Henry's prior existence, and explained that Peter was erased from existence because Henry was never supposed to be born. Instead, Peter was supposed to have children with Olivia and not Fauxlivia, and Henry's existence was an event that changed the timeline too much for the Observers to compensate for, so they opted to erase Peter to solve the problem.("The End of All Things")