Your cost basis will depend on how you acquired the rights. Were they inherited, gifted, or did you purchase them outright? You gain/loss will be computed according to how the rights were acquired.
If you inherited the rights, then the cost basis would be the fair market value on the date that you inherited the rights, general the date of death for the original owner. In this case, you would receive a stepped up basis and gain may be reduced depending on when you sold the rights.
If these rights were gifted to you, which is often the case with mineral rights, it gets a bit more complicated. The basis of gift is the adjusted basis of the donor at the time of the gift. To figure the basis of property you receive as a gift, you must know its adjusted basis to the donor just before it was given to you, its FMV at the time it was given to you, and any gift tax paid on it. See page 9 of IRS pub. 551 for the details.
If you can't calculate basis, then your basis will be zero and the highest amount of gain will be computed. With historical records and some research, a zero basis can be avoided.
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