The Kanji character of Koku translates as "country.' and the character for Ba translates as "Place". The character Koku is pronounced as Koku in Okinawa and as Kuni in mainland Japan. The actual written form, in Kanji, is the same.
The Kokuba family traces their ancestry back almost 2,000 years and the charts to prove it are kept at the home of the head of the family, Koshu Kokuba who is the oldest male of the family line. He is also the former superintendent of the Okinawan School System. The family history goes back to Yoshitsune Minamoto who was the younger brother of Japan's first Shogun Minamoto Yoritomo who became Shogun in 1192. The younger brother was considered a threat to Yoritomo because he was a strong warrior. He was exiled from the court and fled south where his descendants eventually wound up in Okinawa. That is part of their family history.