As a contractor, you pay taxes to any state where you were physically present while working. You also need to pay taxes to your resident state.
If your company was in MN but you only worked in ND, then you will file a ND tax return. As a resident of OH, you will also file an Ohio return. Ohio will give you a credit for any taxes you pay to ND.
Because you were not working in MN, you will not need to file a MN tax return.
Note that as an Ohio resident, if your work in ND was temporary (lasting less than a year), then you can deduct your expenses for travel to the job, including vehicle expenses for all your travel in ND, as well as lodging and meals, unless those expenses were reimbursed.
If you were required to sign a new contract every 6 months or so, then you could still count your work in ND as temporary.
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