Facts:

My dad recently passed away. A few months before this he took out a line of credit on his and my mother's house (with my mother's knowledge). With this money he purchased a house in Michigan and proceeded to quit claim deed it to one of his children. My mother is now having to try and pay the line of credit for a house she does not live in. Does she have a 50% or more interest in the house in Michigan? What legal action can she take to try and recoup some or all of the money used in purchasing the home?

No, if her name or dads name is not on the deed in Michigan, The child who it was deeded to, would own it free and clear if just their name appears on deed.

And if the dad did not put a mortgage on it, then it is just clear.

Why did the dad do this? Was the child who got the house, were they suppose to make payments, was there some verbal agreement.

But if he took one loan, and just bought a house free and clear and gave it away, there is not much you can do, except hope the child that got the free house will help with payments out of the goodness of their heart.