The IRS does not have to award any cost basis your mother can't prove with reliable records.
Assuming your father died before your mother, she inherited his half of the home with a stepped-up basis equal to half the fair market value at the time. You can also prove the amount they paid for the land from county records. You may be able to prove the cost of home improvements made recently.
For example, if your father died in 2015 and the home was worth $200,000 at that time, the cost basis you can prove is $100,000. If the land was originally bought for $5,000, that increases the cost basis you can prove to $102,500. And if she replaced the furnace in 2017 for $4000, now you can prove a cost basis of $106,500.
Note that if the home is in a community property state, your mother inherited a stepped up basis equal to 100% the FMV when your father died, not just half the FMV.
The real estate value on the date your father died can be established by a qualified real estate appraiser.
If your parents divorced and your mother paid out a sum to buy out your father's share (or there was an adjustment in how assets were divided that did the same thing) then that can be used as the cost basis.
Then, as long as your mother was using the home as her main home, the first $250,000 of her capital gains are tax-free. If your father died less than 2 years before the sale, she can include his $250,000 exclusion for a total exclusion of $500,000.
https://www.irs.gov/pub/irs-pdf/p523.pdf
