The dictionary defines the adjective righteous as acting in accord with divine or moral law; free from guilt or sin. Righteousness is the noun derived from righteous.
So what does the motto mean? It means that the life within the land is determined by the right acting ways of the people within the state and depends upon the people doing the right thing. Right actions in personal life as well as political (state) life can sum up the motto.
Historically, the motto was first thought to reflect the beliefs of the kingdom. King Kamehameha is said to have spoken these words in the early 1800's when he presided over what was known as the Kingdom of Hawaii. The motto has been used for more than two hundred years throughout the state of Hawaii on seals, gates and various key places in the Hawaiian Islands.
I believe that this motto did not come out of thin air but was based upon the beliefs of the original settlers of the Hawaiian Islands. It speaks of their relationship with 'Io, the benign and loving creator god. In Daniel Kikawa's book, Perpetuated in Righteousness, he tells of the first brave adventurers who "discovered" Hawaii.
"A great leap of faith was required to continue on into thousands of miles of empty blue on blue.In the northerly direction of this wonderful star, our ancestors searched for and found our new home. We had finally found the place 'Io had provided for us! Here we worshipped 'Io, in this land so much like Atea's first home." (My notes of clarification: Atea is the Hawaiian equivalent of Adam. They were following a star that was spoken of in their ancient prophecies.)Throughout Hawaiian culture there are many things that indicate the desire of the Hawaiians to fulfill their state motto. "Aloha," "ohana" and "hanai" are some of those. See the Afterwards section of Kikawa's book for more details.
Have you observed anything in the Hawaiian culture that indicates the fulfillment of their state motto, "Perpetuated in Righteousness?"