"The Empty Child" is the 9th episode of series 1 of the revived series.
The child's identity was revealed (in "The Doctor Dances", the 2nd episode of the 2-parter) to be Jamie, Nancy's son. She'd pretended he was her brother because she didn't want to admit to being an unmarried mother, in an era when this was considered disgraceful. (From her conversation with the Doctor, it seems she was also under legal age when Jamie was conceived, adding to the potential scandal.)Jamie had been killed during an air raid but the Chula nanogenes from the crashed ambulance ship had repaired him as well as they could. They were not acquainted with humans, however, & mistook his gas mask for part of his body. They then sought to repair other humans similarly, which is what made his condition spread as if it were a plague.
When the nanogenes were able to compare Jamie's genetic material with that of his mother Nancy, they corrected their repairs to all the affected humans & returned them to normal -- or, in some cases, better. One woman who had previously lost a leg found that it had been regrown.
The "empty child" was the term used to describe Jamie in his incorrectly repaired state.