Rose loved him so. This is a true thing and while he didn't seem to be the man that she remembered, she grew to love him even more.
It wasn't easy, at first. She spent many nights waking him from his nightmares and screams and just holding him as he cried softly in her arms. He had seen so much; so many things that a human man wasn't meant to see. Things that the Doctor could handle that this human, made in the Doctor's image, was not equipped to deal with. Still, the human mind is remarkably resilient, maybe even more so than a Time Lord's, and the running sores that were his memories scabbed and eventually healed.
He decided that he needed a proper name. He didn't want "John Smith" or the Time Lord's real, Gallifreyan name (which he had told to Rose one special night). Those were the Time Lord's names, not his. He kept "John" for sentimental reasons, but for a surname, he chose "Who".
It was a joke, you see. His own, private joke. So many times people had asked the Time Lord "Doctor who?" that John just couldn't resist the pun. Doctor John Who it would be.
Their love grew even more. He had the life that the Doctor had yearned for, and he felt the humanity return to him as the mental wounds of the fight with Davros and other past adventures and especially the Time War became distant. He thanked the Creator for his human mind, because unlike a Time Lord's mind, past events began to blur as he grew older. Events became less distinct and he was haunted by them less and less.
One day, over breakfast, he announced matter-of-factly to Rose that he was going to build a time machine. He said this with as much fanfare as any other man might announce that he was going to mow the lawn. For just an instant, Rose forgot who she was talking to and giggled. John gave her that staring, sideways, "just you wait and see, my girl" expression that he had always had. Then he went to work.
It took him years... decades. During that time, they had a child; a son that caused them a minor argument when it came time to name him. Rose wanted to name him "Mickey" and John wanted to call the boy "Alistair". Their compromise was to name him Michael Alistair Who. They were so proud of him, and he of them.
Oh, how John Who loved that boy! To his mom, he was "Mickey". To his father, he was always "Alistair". "So much like my first son," John had said. When Rose pressed him to talk about the Time Lord's son, John had reluctantly told her and they both cried over the young man's fate.
Alistair grew to a strong, intelligent, good young man. He was not, and would never be, as intelligent as his father, but he was something extraordinary and John loved him almost as much as he loved Rose.
Alistair loved his father, too. Others saw Doctor John Who as an eccentric. He puttered around with the strange, blue box he kept in his back yard the way other men his age puttered around in a garden, but Alistair knew his father was brilliant.
When he asked his father or his mother about the time machine being built, he would ask them why it looked like a police box. John would laugh and squeeze Rose's hand and say "Purely because that's what a time machine SHOULD look like, my boy. Trust me; I would know."
When the machine was finished, the Doctor called it Tardis (without the definite article) and he invited his family for a little test drive; a little spin around the past century, he said.
That inaugural flight, as so many things from the Time Lord's life, ended in tragedy. Attemtping to reach 1920's London, the little family instead ended up in the London Blitz of 1939. Exiting Tardis, they were caught in an explosion. When Doctor Who regained consciousness, his wife was gone and the time machine was severely damaged.
John grieved for months. He still had his wonderful son, but his wife, the woman he loved more than life itself, was gone. There was enough of the Time Lord left in him to understand that it was her time and it would have come in 1930's London or their home in 2025. It didn't matter where they had been; it would have happened.
They were stranded in 1939 London, but they survived the war. Alistair became a decorated war hero; a pilot who shot down more German aircraft that anyone else. While John, with Rose's help, had learned to abhor war, he told Alistair that this was the most just war in the history of Earth and he knew it to be true.
Within a few months of landing in 1939, Alistair had met and fallen in love with a young woman named Barbara; a nurse who drove an ambulance. At first, John warned the boy about entanglements. They were travellers in the fourth dimension and there was history to consider. But as the months stretched into a year, John realized that 1940's England didn't have the resources he needed to repair Tardis and he relented. He came to love Barbara as if she were his own.
The war didn't touch them, but the 1950's and a senseless automobile accident took young Alistair and Barbara from him, leaving Doctor Who to care for their children. The eldest daughter, named for her mother, was a joy to the middle-aged Doctor, but the youngest child, who Alistair had allowed his father to name, was his greatest love, outside of Rose herself. Susan and Doctor Who were as close as could be and were inseparable.
By 1965, he even managed to repair his damaged Tardis and in a fit of excitement (as well as the clumsiness of Barbara's boyfriend), Doctor John Who, Barbara, Susan, and Barbara's boyfriend, Ian, were off on an adventure to the planet Skaro. When they returned, they called their secret adventure "Doctor Who and the Daleks".
So when you hear of them, a time travelling very-much-human grandfather named "Doctor Who" and his two grandchildren and a young policeman named Ian, remember where that strange old man really came from... and it will all make sense.