My son is a resident of Hawaii but goes to college in Arizona and works in Arizona. Do we need to file a Hawaii State Tax return?

Whether he has to file in HI or AZ (or both) depends on how much he earned (see the attached chart from the Hawaii Resident Income Tax Forms and Instructions). 

His AZ income is also considered HI income for HI tax purposes. If he has to pay AZ tax on the AZ income, he can get credit against HI tax for what he paid to AZ

As a student attending college out-of-state, he is considered to remain a resident of (i.e. "live in") Hawaii unless he takes action to establish residency in another state (does not have to be the state where he goes to college). 

When you complete the MyInfo (Personal Interview):

1. Select Hawaii  as the state of residence on December 31, 2016;

1. Answer "No" when asked if you "Lived in another state in 2016?" That question only applies if you changed your state of residence in 2016.
2. Answer "Yes" when asked if you "Earned money in another state?". That would be Arizona.

What makes you a resident of a state (or country)?  

Generally, you're a resident of a state (or country) if you intend to either stay there permanently, or return there after a temporary absence. It's where home is – where you come back to after being away on vacation, business trip, overseas or out-of-state employment, or school. Many factors are considered, not the least of which are where you are registered to vote, own homestead property and are licensed to drive.