Having lived close to half of our life in Hawaii, this subject is of interest to us, as we had the good fortune and blessing to study with Hew and his mentor Morrnah Simeona. We feel somewhat qualified to expand on what has been shared already. Morrnah has since passed.
As it was taught to us, Ho'oponopono, to the learned teachers/healers of the Hawaii past, had the meaning of 'to unite the Ohana (family)'. They had the belief that we are a tri-une being and that before anything can be accomplished in this realm by the body, the three beings needed to be united. When a healer was approached with someone with a physical aliment the first line of action would be Ho'oponopono because they figured, rightly so in our opinion, that whatever was going on with the physical, was a disconnect/imbalance in the spiritual/mental body beforehand.
This out-of-balance would also manifest in situations and our interactions with others. They also believed that if these ruptures were not healed in a past life they would carry over and manifest in this present one and present themselves to be healed. I would use this, and still do regularly, when for instance, someone in a store would get close to me once or twice while shopping, I would do a quick Ho'oponopono with the awareness that this entity was sent to me now to forgive, both of us, and make right/heal a past transgression. Like the C's tend to remind us, there is no time as we understand it.
In the long form Ho'oponopono we use the phrase "humbly, humbly, humbly" and we do think that this humbleness is an important mental space to be in when using the process.
"I'm Sorry--Please Forgive me--Thank You---I Love You" PERLOU we expanded on what you posted.
So the question of this post about where this healing method came from, it could very well be, as all the Polynesians say, that they have a lineage to a vast civilization that vanished in the last major cataclysm and could very well have been one of the passed along remembrances.