It should be: I will be calling him tomorrow. Because she says "звонить" which is a future imperfective. If she said я ему позвоню, meaning I will call him, that would denote a single action in the future, but she does not, she denotes a continues action in the future. Or am I missing something here?
"Буду звонить (at a specific time)" - which implies a continuous and rather laborious process of making a telephone call - was a sentence form widely used in the past in everyday speech. Most probably it is because having a call actually go through (esp. long distance) was not at all guaranteed and most likely involved several tries. To make a long distance call in Soviet times one would go to a main post office branch, wait in line at a special window to make a reservation for a specific time, then wait for your name to be called. Then you would hope that the connection would be clear and you'd be able to hear the other party. So much history and heartache contained in one little phrase - "Я буду звонить ему завтра"! But if we said simply: "Я буду звонить ему" (without specifying exactly when you will be calling) - then it can be simply a promise that you won't forget him and will call on a more or less regular basis from time to time.