Can a family member be assigned to provide in-home care for the elderly family member?

Your profile does not give much information. What sort of responsibilities are you talking about? Grocery shopping once a week is one thing. Preparing food so that grandmother has three nutritious meals a day is quite another. Nowadays, most young people have no idea what being a caregiver really means because they did not grow up in an intergenerational household.

How well do you know this grand daughter? Is she responsible and honest? Does she understand what she is getting in to? Depending on the level of care needed this could be quite a job.

Even if her granddaughter is willing you should insist on introducing some outside caregivers anyway, there will be times when g-daughter will need to be away due to illness, vacations or to take care of personal business.

Anyone can be an unpaid caregiver for a relative, there isn't anything that needs to be signed or any criteria. Just a willingness to help out.

When I think of "grand-daughter" I picture a young woman who is just out of high school or in her early 20's. How old is this grand-daughter and would she need to leave her own family and employment?

Would the grandparent be able to pay the grand-daughter? If yes, then an employment agreement would be needed. This would come in handy if in the future the grandparent needs to apply for Medicaid. This shows where some of the money went.