Should I include a cost basis for additional distributions in 2019 on a ISO that was sold in 2018?

Did you have additional sales in 2019? If the ISO was sold in full in 2018, and you have an adjustment in cost basis for that transaction, this should be reported through an amended 2018 return. Can you clarify what you mean by additional distributions if the ISO was sold in a prior tax year?

I purchased/exercised my ISOs in 2017, in 2018 the company was bought out and my stocks were sold. I calculated the cost basis when entering my income from this sell in my 2018 tax return. In the terms of the sale, previous shareholders would get additional payouts/distributions based on how many stocks they owned at the time of the sale. I didn't own (or sold) any of this stock in 2019 but I received the additional payouts. My guess is the cost basis is 0 because I already filed it in 2018 when the actual sell happened. Any insight is much appreciated! I hope that helps clear things up.

I agree, the cost basis is what you spent to acquire the stock, and/or the income you recognized when you took ownership of it. That is what you should have reported in 2018 when you sold it, and it wouldn't change after that.