I lived in Minnesota for 135 days, New York for 103, and Michigan for 54. The rest of the time I was traveling. My parents own a home in Minnesota that I stayed at while I was there. What do I put under 'state of residence'?
I am assuming you are in the My Information section. This address would be for mailing purposes only if the IRS needed to contact you.
As far as determining your state of residency for tax purposes, each state has their own rules. After making this determination you would choose your state (s) of residency for the year also in the "My Information" section see the example below.
Please see TurboTax FAQ - What is my state residency status?