I feel law does not differentiate between first name, middle name or surname. Any change, addition, deletion or modification in name or spellings amounts to name change irrespective of ground realities like marriage, divorce or on numerological considerations.
Name change in school may be easy but may be difficult when you or she will apply for passport, because municipal or birth office records will remain the same. The apt course of action may be going for gazette notification. So visit the concerned office, fill the forms with fee etc, get it published in local dailies and further actions as per their procedure. The procedure on name change may differ from state to state.