The placement of nicht depends on the item you want to negate. Example dialogs:
Elisa: Wir brauchen einen Babysitter für Markus. Claudia und Lydia wollen doch sicher wieder zum Tanzen gehen.
Arnold: Will nicht Claudia den ganzen Abend zu Hause bleiben?
Arnold in unsure whether Claudia or Lydia said to him she wanted to stay at home.
Arnold: Will Claudia nicht den ganzen Abend zu Hause bleiben?
Arnold remembers Claudia said to him she wanted to stay at home the whole evening, but he is unsure if she changed her plans.
Arnold: Will Claudia den ganzen Abend nicht zu Hause bleiben?
The sentence above is grammatically correct but sounds very awkward. If you wanted to negate Claudia stays at home, better say:
Arnold: Will Claudia den ganzen Abend von zu Hause wegbleiben?
Arnold is unsure if Claudia's plan to stay away from home covers the whole evening.
EDIT: I see now, I misunderstood your question. You've wanted to answer the question in the title. But the system stays the same:
Arnold: Will Claudia den ganzen Abend zu Hause bleiben?
Elisa: Nein, nicht Claudia will den ganzen Abend zu Hause bleiben, sondern Lydia.
Elisa corrects Arnold about the person who wanted to stay at home.
Elisa: Nein, Claudia will nicht den ganzen Abend zu Hause bleiben.
Elisa corrects Arnold about the duration of Claudia's planned absence.
Elisa: Nein, Claudia will den ganzen Abend nicht zu Hause bleiben.
Again, this sounds awkward. Better say:
Elisa: Nein, Claudia will den ganzen Abend von zu Hause wegbleiben.
Elisa corrects Arnold about Claudia's general plan for the evening.