The British or possibly the Germans? Someone was going to take them over.
The British and French had come to an agreement between themselves that neither was to attempt to claim the islands for themselves and to respect their independence. Hence why in 1843 when a Royal Navy captain for some reason got above himself and demanded, successfully, that Kamehameha III cede his kingdom to Britain as soon as the local commanding rear-admiral found our about it and arrived on the scene he told him to wind his neck and restored the monarchy and apologised as fast as possible. Now that's not to say that things couldn't change in the forty odd years until the monarchy was overthrown but it suggests they weren't going out of their way to target the kingdom.
My usual suggestion for this scenario is to look at what happened with Tonga as a protected state - the UK was responsible for foreign relations and defence matters with the kingdom retaining freedom on internal affairs. Since the big attraction of Hawaii at the time would be their location and the great harbour at Pearl Harbor then one solution is for an outside country to offer the Hawaiians protection in return for the right to build a naval base and station some troops there. Whether that could be the UK, France, the US, Russia or whoever is wide open.
One thing to bear in mind is what continued independence could do to the economy. If someone other than the US moves in and takes influence over the islands then the Reciprocity Treaty of 1875 could very likely not happen. Since the US after the treaty was a major market for their exports is there anyone else in the region that could be substituted? One slightly off the wall idea I had was for the Hawaiians to barter military access to Pearl Harbor to multiple countries in return for recognition of their independence and trade agreements as a way of balancing them out.