When we sell her home, what should be do with the proceeds?

Medicare is an entitlement program that is not based on need. So whether she has the house or cash will not affect her Medicare benefits. Medicare is health insurance that pays for hospital and skilled medical care, e.g. doctor, nurse, physical therapist.

Medicaid is based on need. Each state sets a maximum resource level that a claimant is allowed to have. This is usually less than $15,000, often as low as $2,000. If the claimant has more cash than her state's allowance, she will become ineligible for Medicaid. Since Medicaid has a 5 year look back period, there is no point in transferring the proceeds from the house sale to someone else. The exception is if the house proceeds will not be spent down within 5 years. Then whatever is transferred out of her name now will be protected after 5 years.

Medicaid is a safety net. It is for people with very low assets.