I need to use some sort of component to boil water inside of a container that will be triggered by an arduino signal. How?
You should confine yourself to materials that are safe for contact with potable water, which rules out a bunch of possibilities. 

One idea might be to cannibalize a rice cooker- they have a heater below the rice bowl and a spring-loaded thermostat. I've seen them as cheap as $8 at discount stores, but $15 might be more typical. They are designed to shut off when the liquid is gone and the temperature rises. Perhaps you could substitute your own (open) container for the rice bowl. 

I do hope you're not planning to make a homemade pressurized boiler. If so, I will have nothing to do with it, and you should check your local laws and regulations about what kind of testing and approvals may be necessary.