New Zealand Herald editor-in-chief Tim Murphy, who was on board the flight, tweeted that the takeoff was aborted "at speed mid-runway".
A light flashing in the cockpit had alerted the pilots to a potential problem, but Air New Zealand later said it was a fault with an indication system, not the engine.
Passengers were put up in hotels for the night as a replacement part was sourced.
But as they waited in the airport for the rescheduled flight on Sunday night local time (Monday night NZT), engineers still hadn't been able to fix the indicator.
Mr Murphy complained of miserable handling by the airline and "queuing perpetually and feeling like Tom Hanks living in the terminal".
Trade Me founder Sam Morgan, also among those stranded, tweeted of "the pain of 48 hours stuck in airport limbo in the tropics with kids".
Air New Zealand flew the replacement part from mainland US to Hawaii but passengers were sent back to their hotels for a second night.
Mr Morgan tweeted on Monday morning: "Last night @FlyAirNZ checked us all in (3 hours) only to find out the part didn't work. This morning we're off to do the same thing.
"Great if they could fix the plane BEFORE sending us to the airport for 3rd attempt. #nz9"
But as the ever more frustrated passengers - now with "grumpy" children, some reportedly vomiting from a lack of sleep - began to arrive at the airport for the third day, they were told of a further delay.
That meant four more hours in the departure lounge for those unable to score a seat on a Hawaiian Airlines flight to Auckland. About half boarded flight HA445 at 2.40pm on Monday, leaving behind increasingly "anxious and annoyed" passengers.
Only those flying economy were left waiting in the airport, a passenger, who didn't want to give her name, said. She said the situation had brought her "pain and discomfort".
"There has been very little help from the airline," she said.
As the delays continued, anger grew: "NZ9 is turning into a complete farce. Farce is the politest word I could use," Paul Dolley tweeted.
Passenger Paul Cavanagh said at 5pm on Monday, one hour before the Air New Zealand flight, now operating as NZ6889, was due to depart, the plane was "nowhere near the gate". The flight was "not even on the departure board any longer", he said, calling the situation "beyond a joke".
Minutes later, passengers were told the flight had been rescheduled yet again - until 7.30am the next day (5.30am today NZT).
People understood about the technical fault, he said, but were angry at the lack of information.
"We've had little communication, and this morning the communication to get on the bus was pure luck as it was word of mouth from passengers. There were no text messages, or emails, or phone call, just a memo at the front desk of the hotel," Mr Cavanagh said.
Passenger Phil Wardale said people were "angry [Air New Zealand] took us from the hotels before they knew" if the flight would go ahead.
In a statement Air New Zealand said the replacement part had failed "to resolve the engineering issue as expected".
Another replacement part was being flown to Honolulu, with engineers on standby to fit it.
Last night the airline issued an "unreserved apology" to all passengers, admitting it had "repeatedly let them down", and offering $1000 compensation in cash or airpoints. "We are incredibly sorry to have let our customers down and will conduct a full review to see how we could have managed this disruption better."
