Of the two choices offered, that which leaves out in seems preferable. However, if "of the next day" is left off, include the word in, or otherwise reword:
She told her to phone him in the morning.
She told her to phone him the next day.
The former of these phrases may be ambiguous regarding if the morning is when she told her, vs. when the call is to be made. The original choices have the same ambiguity if the narration is completely past tense. Was it on the morning of the next day after some other event that she told her to phone him, or did she tell someone, "Phone him in the morning"? The ambiguity can be avoided by using a direct quote.
Edit (after context was added to question): One might convert «Mrs. X: "Phone me tomorrow afternoon."» to one of
Mrs X. said to phone during the afternoon of the next day.
Mrs X. said to phone during the next afternoon.
Mrs X. said to phone the next afternoon.
Mrs X. said to phone in the afternoon.
Phrase "the next day" would be left off if context makes it clear. "During" can be used as shown and cannot be replaced by in.