The overhead associated with operating a commercial nuclear power plant (maintenance, safety requirements, fuel transport and storage, etc) means they don't become economically viable until you're servicing a population of about a half million. That's what Honolulu doesn't have a nuclear plant even though it'd be almost ideal for their remote location. Currently they get most of their electricity from burning fuel oil, and consequently have the highest electricity prices in the U.S. - about $0.30/kWh vs the national average of $0.12/kWh. Cost on the islands other than Oahu is even higher (about $0.45/kWh) because they have less access to oil and have to rely more on renewables.
With a population of just under 400,000, you couldn't run a small commercial reactor full-power 24/7 as they like to be run. You'd have to ramp it up and down throughout the day, which greatly increases operational costs. In the rest of the country, nuclear provides 24/7 baseline power. Coal plants can ramp up/down more quickly, but it still takes a while so they also provide baseline power. Fluctuations in power use through the day are handled by oil and gas plants (which can ramp up/down almost instantly) and hydro (which can ramp up/down instantly).
A RTG (generates heat through nuclear decay, not an induced nuclear reaction) could work. The Soviets used to power many of their remote lighthouses with them. But the wind in Antarctica is very strong and very consistent, and would seem to be the obvious go-to energy source given the scale and remote location (minimal maintenance crew).