Can my mother, who is currently on Medicare, keep the money if she sells her house?

Eligibility for Medicaredoes not depend on a person's income or assets. Anyone who qualifies because of age (65 or over) or disability (24 months qualified for Social Security disability See also:
Medicare Coverage for Annual Physicals benefits) is eligible for all four parts of Medicare, regardless of how much money he or she has. So, your mother selling her house would not affect her Medicare eligibility in any way.

On the other hand, selling her house might have an effect on her eligibility for Medicaid, if she applies for it to supplement her Medicare coverage. Medicaid is the federal government's main medical coverage program for people with very low income and few assets. In counting a person's assets, Medicaid does not consider the value of a house if the person applying for Medicaid is living in that house. Once your mother sells the house, whatever money she keeps from the sale would then count as her assets for purposes of Medicaid eligibility.