Say, a mother is filling out a form for her daughter (patient) at the hospital. Which one should she put down in the given space - Mother or Daughter? I am assuming either one of these must be the answer since there is a colon after the of. Also, does it have to be A mother to feel like the sentence is complete. Anyway, my mother actually wrote "I am her mother" and told me to calm down... and I did, but I could always ask here on StackExchange.
This is why my first grade teacher taught me to answer questions with complete sentences. Your mother's answer was a complete sentence, and resolved the ambiguity.
"I am her mother" is correct.
Based on the phrasing of the question in the title, "mother" would be the correct one-word answer. The question is phrased in terms of "(I am the person filling out the form.) My relationship to the patient is that (I am the patient's): ____".
The word "my" determines the implied subjects of the implied sentences and clauses.
If the patient were filling out the form, the correct one-word answer would be "self" or "me" or "myself".
"My relationship to the patient is mother." If A is the mother of B, then the relationship of A to B is mother. If the mother was the patient and the child was filling out the form, then she would say "My relationship to the patient is daughter" or "child".