Geography, Geography, Geography.
Oahu, and specifically Honolulu and Waikiki offered well formed, almost perfect natural harbors for both a new centralized government under King Kanehameha, but as with many new and forming kingdoms over the years, in an almost feudal sense, the capitol was simply wherever the king was at the time. And thus when he was in residence at his lodge on Waikiki the capital was too. It would rotate to other places during his reign, and that of his son Kamehameha III who after his older brothers death from illness n a state visit to London would reign for nearly 30 years(most of it under his majority) and do much to formalize the state and build upon the foundations of his brother and father.
He undertook to formally move the capitol full time from Lahaina, on Maui which had hosted it for much of the past two decades when he made the move in the mid 1840's. The new capitol of Honolulu was simply a major improvement on the other contenders. It had a large natural harbor, which wasn't unique to be sure around the islands, but it was also southward facing, making it that much better from a quality of life focus. While Oahu itself was also separating itself from the other islands as a major economic driver for agriculture and shipping with foreign companies, other concerns like whaling ship traffic could also prop up other ports or simply be attracted to a new port of call that was intentionally being developed. This also brought new people to the island, most notable Japanese and other Asian immigrant populations either as crew of ships or as cheap labor.
Its also worth noting that by the time of the overthrow of the Kingdom and direct US rule in the 1890's, while Honolulu and Oahu had separated themselves, it wasn't irreversible and by degrees(if large ones).
The final driver was just to the West of the city, in one of the finest natural harbors on the planet. That of course being Pearl Harbor. Almost from the first time it was discovered by European sailors its possibility as a fleet anchorage protected from weather and enemy action, along with ease of loading was appreciated. But large coral reefs and tight channels made any task for the moment too much.
It wasn't until 1899 that a permanent US Navy presence was established even in Honolulu with a few service craft and small shore facilities to service and coal vessels on their way to the Philippines and the Asiatic Squadron. But surveying work and excavation and dredging began soon after.
By the 1910's there are always a few naval vessels present, along with a growing Army garrison. And in the interwar years the island routinely hosts the entire combined US fleet for summer for Fleet Problems. And while not quite ready to serve as the home of the largest sea going organization in the history of man, by the 1930's Pearl, and nearby Hickam Field and the other military installations around the island had reshaped to island in a major way, including driving even more demographic changes.
Looking at census numbers it easy to see how the increasing US commercial, and military presence in the islands centered on Oahu quickly push its growth. In 1900 there were 40k people in Honolulu, by 1940 that had become 140k! There was simply no way for the other islands and their population centers like Hilo to keep up.