When the boy was born she knew he would never meet his father and she didn't want him so she named him and left him at the orphanage.
The boy was called Kahn after his father's real name and just left a note saying his name.

*a few years in the future*

The boy was progressing very quickly. Quicker than most other children of his age. That is when they decided to take him for bloods. To check the boy was human. If not he will be handed over to the enterprise.
They took enough blood to run the tests and then said they would be back with the results soon. What they found they didn't like. Khan was sat outside the office whilst his social worker got his results. Human augment he belonged to the enterprise now he had been revealed as the only one of his species awake the rest were in sleep.
The enterprise was very tempted to wake John Harrison to see if he recognised his own son but they decided against it after the destruction his father caused last time. It was interesting to see the boy he was a teenager and was so strong. He had been kicked out of school after a boy hit him and when he hit back the boy had to go to hospital. The enterprise wanted to put him to sleep seeing how dangerous he was to others. He definitely had his father's anger issues.
The boy was starting to look like his father but they really wanted to know who his mother is. They looked at the blood test results and found a match two both parents. Khan/John Harrison and a Molly Hooper.
They recognised the name and soon found her. All they had to do now was remind her of Khan/John Harrison and her son.