The Bessants, of Browns Bay, are trying to raise the $375,000 it will cost for Jesse to undergo a radical new treatment being trialled at a Texas clinic.
Jesse's ganglioglioma tumour was diagnosed last year after his head developed a tilt, one eye began flickering and he had trouble with his balance.
Amazingly, after undergoing a biopsy in September, in which a small sample of the tumour was removed for testing, his condition improved greatly.
"Just taking that tiny piece away relieved the pressure. It was just pure luck, but it goes to show, a millimetre's growth is going to put him back to how he was," Mr Bessant said.
But it is only a matter of time before the tumour grows more.
New Zealand doctors suggested treating Jesse with chemotherapy, although this has been shown to be largely ineffective against this sort of brain tumour.
Then a family member alerted them to a cancer treatment facility in Houston called the Burzynski Clinic which is carrying out trials of a new therapy.
The treatment is called antineoplaston therapy, which has nearly completed its second stage of Food and Drug Administration-approved trials.
It involves injecting patients with synthetic chemicals called antineoplastons, intended to protect the body from disease.
"We were just gobsmacked," Mrs Bessant said.
"People with no hope, who had high-grade tumours and had been told to go to hospice were being treated there and were surviving.
"We got in contact with some patients who've actually got the same tumour as Jesse who are doing the treatment and their tumours are shrinking."
Mrs Bessant said Jesse would have to get a catheter port installed directly into his heart, and he would have to wear a backpack containing the drugs and a pump for around 23 hours a day.
The full treatment could take more than two years and cost $375,000.
"But once we hit $100,000 for the initial procedure and consultation we know that we'll be able to go over and do the treatment and that'll give us three months supply of drugs as well."
Mr and Mrs Bessant would be trained to administer the drugs themselves so they could return to New Zealand after four to eight weeks.
Jesse's New Zealand doctors had mixed opinions about the treatment, but they would support the family when they came home, Mr Bessant said. "We just want to give him the best chance we can. This is really, as far as I'm concerned, his only hope. It's his only hope of living."
HOW TO HELP
People who want to donate money to help Jesse can visit his website: www.jessebessant.co.nz or www.teamjesse.co.nz
A trust account has also been set up for donations: Jesse Noah Bessant, ANZ Browns Bay 01-0121-0211912-30