What is the state tax filing for a dependent full time college student who earned all their income via an internship in a different state than their residence/parents?

The student would file a non-resident return in Illinois if required to based on her income. Here is a link to the Illinois Revenue Department that you may find helpful:  Illinois Dept of Revenue

The general rule is: your report all your income on your home state return, even the income earned out of state. You file a non-resident state return for the state you worked in and pay tax to that state. Your home state will give you a credit, or partial credit, for what you paid the non-resident state. You will have to file a non resident IL state return and pay IL tax on the income earned there.. You will also file a OH full year resident return and calculate tax on ALL your income. OH will give you a credit, or partial credit, for the tax you pay IL. So, there will be little or no double taxation, but you have the cost and hassle of filing two state returns. Do the nonresident state return first.

If the OH resident earned income in a reciprocal state (IN, MI, PA, WV. KY) no tax is due to the non resident state. It would only be necessary to file a return to get a refund of mistaken withholding.