"They paid back 1/3 in the first year, which is what would've been taxable, so there are no taxes due in the first year."

But you didn't already submit for 2020 taxes, then repay and amend 2020.

"If someone takes a 401k distribution but then immediately pays a portion of it back is the payback amount tax-exempt to NJ and where do I report that on the NJ-1040?"

NJ doesn't tax rollovers:

"If you roll over a lump-sum distribution from an IRA or a qualified
employee pension or annuity plan into an IRA or other
eligible plan, do not report the rollover on line 20a or 20b if
it qualifies for deferral for federal tax purposes."

Let's understand that a covid disaster distribution was qualified as able to be rolled over and would be treated as if that was trusee-to-trustee and given three years.

So the Math is Net. Net reportable.

That's why each repayment means you file Amended for the year affected where what should be reported is now changed, because there is a three-year "rollover" equivalent treatment.

If you pay it all back in 2020, there is nothing to report. If you pay some of it back in 2020, the rest carries to the three year treatment. If you repay some in 2021, you can amend 2020 to show the Net taxable has changed. If you repay it all by 2022, you amend 2020 and 2021, because now, none of it is taxable, as if it never happened, as a lookback function. If you repay none in 2020, there is tax on that third. If you repay some in 2021, but not any further amount, you would amend 2020 and also file for how it impacts the amount carried into 2021, and then the final amount is taxable in 2022.