US influence in Hawaii did not start until 1900, with admission into the Union in 1959. Before then the Polynesian culture of Hawaii was ruled by a succession of kings and queens starting with Kamehameha I in 1795. British influence began in 1778 with the discovery of the islands by James Cook.
King Kalakaua (1874 to 1891) issued a series of coins including 5 cents, 10 cents, 1/8 dollar, 1/4 dollar, 1/2 dollar, and dollar in 1883. This page does not apply to the 5 and 10 cent coins. You can find the denomination of the dollars on the reverse side (the side with the shield) using 1/8 D, 1/4 D, 1/2 D, and 1 D annotations. Some of these coins were issued in silver, others were issued in copper. All genuine coins are valuable.
The 1/8 dollar coins are extremely rare, cataloging well over $20,000 US dollars. There were only 38 coins produced. Hope you have one!
The other dollar coins are quite valuable as well. If you have a genuine coin, you will be pleased with these values. Here is a listing of approximate values from the coin catalogs.
worn: $40 US dollars approximate catalog value
average circulated: $80
well preserved: $150
fully uncirculated: $300
Values above apply to genuine silver coins; genuine copper coins catalog near $18,000
worn: $100 US dollars approximate catalog value
average circulated: $200
well preserved: $300
fully uncirculated: $1000
Values above apply to genuine silver coins; genuine copper coins catalog near $10,000
worn: $300 US dollars approximate catalog value
average circulated: $500
well preserved: $800
fully uncirculated: $4000
Values above apply to genuine silver coins; genuine copper coins catalog near $15,000
Since this coin is so valuable, and since modern tourists flood the islands every year, there is strong pressure to produce counterfeit coins. Some counterfeits are of superb quality, produced by well-financed thieves. Most are low quality and easily distinguished from genuine coins. If you think you have a genuine coin, you would do well to have it authenticated, graded, and encapsulated by one of these services: PCGS, NGC, ICG, ANACS. Look them up on the Internet. Do not use other services.
The image below shows a real silver coin next to a fake copper coin. Most everyone can see the marked differences between the two. Genuine copper coins exist, but this one is a counterfeit.