1: Mom asked Gramma why she hasn't been answering
2: Mom asked Gramma why she hadn't been answering
3: Mom asked Gramma why she didn't answer
In both #1 and #2, the implication is that Gramma has repeatedly failed to answer. But #1 further implies she was still ignoring the phone right up until when Mom asked why (or, noting StoneyB's comments below, up until when the statement was uttered/written).
In #2 it's at least possible she simply failed to answer last Wednesday, say, (when Mom called several times). Or again, as StoneyB says, continuously, right up until she was asked about it.
In #3 there isn't even any implication of how often Gramma ignored the phone. It could have been repeatedly, but it's possible Mom is only asking about one call she made at some time in the past.
Note that in OP's example, Mom asks Gramma about failing to answer the telephone (presumably, when Mom called). But in my "cut-down" versions she could be asking about Gramma failing to answer a question (that Mom just asked). That admits of several more temporal interrelationships (1a: why she isn't answering, 2a: why she hasn't been answering, 3a: why she doesn't answer).
Thus it's largely a matter of style/emphasis. There are different possible implications, but several meanings could apply to several verb tenses.