His daughter Elijah wrote she "woke up with so many messages ... Thank you all for the support, strangers, friends, family and many more, thank you."
Mr Paterson had been working in Cairns and returned to Napier to see his wife and four children every three or four months.
Queensland police said he disappeared but was never reported missing. However, seven days ago, detectives began a search for him and, on Thursday, their suspicions of foul play were confirmed.
A public tip-off led them to an area of thick bushland 100m from the nearest road, where they found his body which had been "cut up" and stuffed inside a chilly bin. District Duty Officer John Rhodes said the remains were "found on a stretch that leads off to one of our community missions, it's not travelled heavily, there would maybe be five cars a day."
Police believe Mr Paterson was stabbed to death.
David Leslie Hickson, 32, was arrested early on Thursday morning and charged with Mr Paterson's murder as well as interference with a corpse, after police executed a search warrant of his home. The crimes were alleged to have occurred on November 7.
Hickson was remanded in custody after appearing in Cairns Magistrates' Court on Thursday. He is set to appear in court again on December 1.
Last night at 9pm (New Zealand time) police completed a forensic investigation of the primary crime scene - a single-storey rental house in the southern Cairns suburb of Edmonton.
Two scene guards had been protecting the property 24 hours a day for almost a week, while crime and forensic officers worked to uncover evidence. A dog squad and four detectives also scoured the property looking for any sign of a weapon.
Mr Rhodes said: "Forensic samples taken from the house will be sent to Brisbane for processing, and there are still witnesses we're talking to just to tie everything together.
"There's going to be a full search of the house, which hasn't been done because they didn't want to contaminate the scene, it will be a search to find any other evidence."
A doorknock of residents who lived nearby had not turned up any new information or leads.
The result of DNA tests were not yet available and it was not clear when the body would be released back to Mr Paterson's family in New Zealand.
Whanau declined comment.