The Honolulu Cloning Technique is a method developed in 1998 at the University of Hawaii that uses adult cells and nuclei for cloning mice in multiple generations. It involves removing the nucleus from an egg cell, injecting a donor nucleus, and chemically stimulating cell growth to create embryos that are then implanted in surrogate mothers. This technique was successfully used to produce three generations of cloned mice and had benefits like a higher success rate than other methods and providing insights into egg reprogramming. However, it also had risks like trial and error causing mutations initially and reducing genetic diversity.