My son goes to college in Ohio. Last summer he had a paid internship in Washington State(no state taxes). Is he required to file state return and maybe pay taxes in Ohio?

In what state is your son a permanent resident?  If you claim him as a dependent, then college is a temporary absence from his permanent home (your state).  He must declare ALL income on his resident state income tax return, regardless of where earned.

He is a resident of Ohio. I cannot claim him as dependent because the income he earned through his internship counts for over 50% of his annual living expenses. Thank you

Then all his income is to be reported on his resident state income tax return.

Thank you