He also urged the former NSA contractor not to leak any further documents or commit any acts that could be considered treason.
In a plea to his son, Lonnie Snowden said: "We love you and we want you to come home. We want you to be safe, we want you to be happy. But I know you're your own man, and you're going to do what you feel that you have to do.
"I believe firmly that you are a man of principle, I believe in your character. I don't know what you've seen, but I just ask that you measure what you're going to do and not release any more information".
National Security Agency (NSA) whistleblower Edward Snowden claimed President Barack Obama "closed the door on investigating systematic violations of law" and "deepened and expanded several abusive programmes".
Snowden said during a live chat on the Guardian website that he waited to release documents until after Obama became President because his election campaign "gave me faith that he would lead us toward fixing the problems".
He said it was seeing a "continuing litany of lies" from senior officials to Congress that led him to leak the classified documents.
Edward Snowden, the former National Security Agency (NSA) contractor who exposed the US Government's secret phone and internet surveillance programmes, said the US "is not going be able to cover this up".
He said that by openly declaring him guilty of treason, the US Government had "destroyed" any possibility of a fair trial in the United States.
"The US Government, just as they did with other whistleblowers, immediately and predictably destroyed any possibility of a fair trial at home, openly declaring me guilty of treason and that the disclosure of secret, criminal and even unconstitutional acts is an unforgivable crime", he wrote.
"That is not justice ... I did not reveal any US operations against legitimate military targets," Snowden continued. "I pointed out where the NSA has hacked civilian infrastructure such as universities, hospitals and private businesses because it is dangerous".