He was found in the Waikato River three days later.
After a coronial inquest in June last year, Coroner Wallace Bain ruled in December the 21-year-old took his own life "in the precise manner and place he previously said he would".
Bain said Nicky's death could have been avoided if his parents had been listened to and that the young man's care fell "well short of what he and his parents would have expected".
When the whānau asked for an apology and compensation, Waikato DHB interim chief executive Derek Wright and board chairwoman Sally Webb said the DHB's insurers had complained to the Solicitor-General about "procedural concerns" during the inquest by Bain.
However, Stevens said to find out now from Crown Law that the DHB wanted a new coroner appointed had left the family "gobsmacked".
"It's like we have continually been punished for standing up for our son, for standing up for the truth, and for standing up for accountability, and every step of the way we've paid a huge price for that."
She said if the family had to endure another inquest it would traumatise them.
"It's been drawn out for nearly four years and then the whole process was turned into this very adversarial process by the DHB lawyers and it's not supposed to be that, and this is just a continuation of that.
"There's no respect for the process, there's no respect for the family, there's no respect for our dead son.
"There's no support. The DHB are prepared to pour huge amounts of public money at this, overturning the coroner's court simply because they refuse to accept any accountability."
The request came from counsel for the DHB, Paul White, and Harry Waalkens, acting for the psychiatrist in charge of Nicky's care at the time of his death, who cannot be identified.
When asked to confirm whether the DHB was seeking a new coroner's inquest a spokeswoman would not say.
She referred to a previous statement by the DHB which said they did not make the decision lightly and understood it had already been a "lengthy and painful process for the family".
"As per the Coroner's Act, we were sent the Coroner's draft report to provide feedback on. We responded with a considerable number of procedural concerns, not all of these have been acknowledged in the final report," the statement said.
"Our insurer has instructed the DHB's external counsel to raise concerns about the coroner's process with the Solicitor General, with our support."
In the letter sent to the family, dated today, Crown counsel Bronagh McKenna said: "The Coroners Act 2006 (the Act) provides for a process whereby, if satisfied that a further inquiry should be conducted (for reasons including rejection of evidence or irregularity of proceedings), the Solicitor-General or the High Court may order another to be opened".
The whānau wrote to Prime Minister Jacinda Ardern on Waitangi Day asking her to intervene in the case to appoint an independent mediator to assess a fair compensation. They received a letter of acknowledgement from her office.
Stevens told the Herald previously that compensation was not just about money but instead a sincere apology from the DHB that it accepted responsibility for Nicky's death.
"This is about principle and closure and actually making a change for families that have been affected, and setting a precedent," she said.
"We want a precedent for better accountability and support for families when things go wrong."
