Her son said there had been a failure in "bog-standard policing".
"We just don't want this to happen to anyone else," he said. "Without changes, other families are going to lose loved ones, other people are needlessly going to die."
Mr Warren told the BBC his sister had sent "incoherent and disorganised" text messages on the morning she went missing. In them, she said she had been on the phone to Barack Obama and that she believed she was being chased by the FBI.
'Tragedy'
The inquest jury, which returned a narrative verdict, concluded she had been suffering from a mental disorder.
Her family believes her death was an accident.
"It was absolutely not suicide," Ms Barnes said.
Met Police Ch Supt Colin Wingrove described the death as a "tragedy".
"It is important we acknowledge the findings of the inquest," he said.
"I hope to be able to apologise in person to the family for the identified shortcomings in the police response."
He added: "The Met fully respects the finding of the jury and will seek to learn any lessons that may come from the hearing."