It doesn't matter where he came from or who's son he was.
All that matters is that he was always just being there.
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He's Humanity Personified.
My password is password

it's NOT his son. Chauncey might have been the son of a friend who died and the rich man took him in, feeling sorry for him. if he was his son, the black lady would have brought that up, and Chauncey would have had some money coming to him.
www.thefearmakers.blogspot.com

Another clue is how well the old man's suit fits him, suggesting he inherited his physique.

The implication was that the attorney believed he was being played by Chauncey when they first met. This is evident in his discussion with the female attorney, when he states he "was played for a fool" and "any chance of a political career is over". He believes Chauncey is some sort of operative between the "Old Man", Ben Rand, and the government. The Doctor reinforces this in the arranged meeting with the attorney, when he senses the attorney's suspicions that Chauncey is an important person. The Doctor just plays along, and tells the attorney to "not speak of this to anyone".
It's obvious the attorney thinks it would be to his detriment to go to the media concerning his encounter with Chauncey. However, the film leaves the impression that the attorney is curious, and would like to learn more. The film doesn't develop this subplot any further. The meeting with the doctor is the attorney's final scene.

Yesmy assessment as well
I haven't read the novel so don't know if there is more precise info there but the old man was not the type of person to take in a foundling and allow him to try on his clothes w/o some relationship aspect I would think
Maybe illegitimate childmaybe had relationship w/earlier version of a housekeeper who died at childbirth
And difficult to believe that the lawyer who wanted him to sign the release didn't recognize him from all news reports and come forward to get his own 5 min of famehe obviously knew Chance the gardener and Chauncy Gardner were the same person
He was wearing the same suit for gosh sakesthe housekeeper recognized him right off
"That's the beauty of argument, Joeyif you argue correctly, you are never wrong"

I also always assumed that Chance was the old man's son. It wasn't that unusual to keep mentally retarded or mentally ill children hidden away from the world back in the day.
It's also implied that while the old man felt enough responsibility towards Chance to have him cared for at home instead of institutionalized, he never treated Chance like a son. When the lawyer asks Chance whether the deceased is a relative, Chance pauses and says "I don't think so" rather than "yes" or "no." Now, granted, Chance didn't know or understand much of anything, but even he would have grasped the concept of father/son if he had been told about it, and to speculate about it if he hadn't.