Context Hawaii at the time was little more than a puppet state of the US. The US had forced the monarchy in 1887 to hand over Pearl Harbor as a lease. Consequently, it had then a strong military presence right next to the Hawaiian capital. Hawaii on the other hand did not have any military to speak of. What is more, the Hawaiian economy and the political circles were also controlled by mainly American expats and local pro-annexation advocates such as Sanford Dole.
The coup The coup leaders had strong ties to the American envoy, John Stevens. The queen was probably trying to avoid escalating the situation and antagonizing the Americans. However, if she thought that not arresting the coup leaders would appease the Americans, she was wrong. Shortly after, US marines from USS Boston occupied strategic positions in Honolulu, the queen abdicated and the royalists peacefully handed over the government to Sanford Dole, the local pro-annexation advocate, who then served as President of Hawaii until it was formally annexed by the US.
Hawaiian society in the 19th century Please note that while the local pro-annexation leaders were American expats and descendants of American missionaries (like Sanford Dole), anti-annexation advocates were actually British and other expats (and descendants of British expats). At the time of the coup, Hawaii practically already had a colonialist social structure. Little was left of original Hawaiian culture in the upper echelons of society, the economy, the army, the government, parliament, the judiciary etc were controlled by Europeans. Further, the pro-annexation party had managed to disenfranchise other parts of the population (people of Asian descent for instance) with racist provisions in the 1887 constitution (the Bayonet constitution) that had been written by pro-annexation advocate Lorrin Thurston. Native Hawaiian society had so far failed to adapt to the new reality and was completely unable to deal with the situation or exert any kind of influence. They had already lost control of their country when they bowed to the moral authority of Western missionaries and Western ship captains and adventurers one or two or three generations earlier.
Could they have preserved their independence? I do not see how. Certainly no longer in 1893. Some traditional states managed to play our several colonial powers against one another, notably Siam. But compared to Hawaii before the arrival of Europeans, Siam was a country with developed institutions, a strong military, a long history, and strong international relations. They might conceivably have succeeded in regaining their independence later (like Tonga) instead of becoming an area of European settlement and a US state had they cultivated closer ties to the British or the French instead of the Americans.