A 47-year-old man was restrained by locals at the scene and police later arrested him on suspicion of attempted murder and terrorism offences.
The terror probe led investigators to the Welsh capital Cardiff, where they searched a property said by media to be the home of Darren Osborne.
"I'm not going to defend him, but he's my son and it's a awful, bad shock", she told ITV.
Security Minister Ben Wallace said the man was not known to the security services and police said they believed he was acting alone.
British newspapers The Guardian and Daily Mirror identified him as Darren Osborne, who lives with his partner and four children in Cardiff, Wales.
"The attack unfolded as a man was already receiving first aid at the scene, sadly that man has died", Mr Basu said earlier.
They praised Imam Mohammed Mahmoud as an model after he defended the suspect when he was set upon by a group of angry locals following the attack.
Earlier this month, a van veered into pedestrians on London Bridge, setting off vehicle and knife attacks that killed eight people and wounded many others on the bridge and in the nearby Borough Market area.
According to witnesses, the driver left the vehicle, laughed and shouted, "I'm going to kill all Muslims. He did what he did deliberately to hit and kill as many Muslims as possible, so he is a terrorist".
Police said the 48-year-old man who was driving the auto has been arrested and taken to a hospital as a precaution.
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He said there had been a spike in Islamophobic hate crimes since the London Bridge terrorist attack.
He said on behalf of the family: "We are massively shocked; it's unbelievable, it still hasn't really sunk in".
Police arrested the van's driver, but were trying to determine whether the collision was accidental or deliberate.
Video posted on social media appears to show police and bystanders surrounding an injured person on the ground after a van ploughed into worshippers near a London mosque.
London's police commander Cressida Dick said the attack was clearly aimed at Muslims.
Rabbi Herschel Gluck said the incident hurt and was an attack on "every single Muslim in the United Kingdom and beyond".
"He has a fracture in his skull - but they still don't know why the bleeding from his ear is not stopping yet", the Somali-born man said.
He told the crowd "an attack on one faith is an attack on us all".
Flowers have been laid near the scene, with one card reading: "This is an attack on all Londoners and on my community".
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