Your basis will be the value of the rights when your mother passed, assuming the rights were part of her estate. If you decide to sell, you’ll owe between 0 and 20% on the gain in federal taxes plus state taxes, if applicable. The gain is the amount you sell it for, less your basis.
It may be tricky to value these rights since they are not producing. You may can use the lease bonus as a starting point. Pecan Tree performs mineral valuations. You can find their website here or in the page directory at the top banner under the mineral services tab.