Depending on your income, you won't owe an underpayment penalty if your total payments into the system are more than 90%, or more than 110%, of last year's tax liability.
If you think the tax (15% of the gain) could be substantial, you will want to make sure your estimated payments add up to at least 110% of last year's tax liability. You could pay more, but that's the minimum to be safe from a penalty.
Documenting cost basis on a gift of a house like this is one of the reasons that simply giving a house to a child is considered poor estate planning. There were other ways to make sure you got the house (and could take care of it if your mother became ill) that might have cost more up front in attorney fees to set up, but would save you trouble and maybe taxes down the road.
Since you have to deal with the situation as it is, make the best effort you can to fairly document the basis, and hope you don't get audited.
