What kind of event would cause the current Mitochondrial Eve to be replaced by a new one?
One possible scenario would be that a particular woman develops a mutation rendering her much more 'fit' (in the evolutionary sense) than her peers. For the sake of argument, lets take the very simplistic case of an X-linked recessive mutation causing the women who bear it to have say 10 children at each birth. If this mutation is then transferred to the offspring, it will very quickly spread across the entire population. 

If this continues for a few generations, the crushing majority of the earth's population will be the descendants of that one woman who carried the original mutation, she would be the new mitochondrial Eve.

I want to stress that this is nothing but a very simplistic thought experiment. Such a mutation is i) very unlikely, probably impossible, ii) likely to render the carrying individual less fit rather than more as each birth would be exhausting, increasing the chances of dying during child birth.



Update:

I have specified that the mutation has to be X-linked and recessive. This means that it will only present a phenotype in women who are homozygous for it, women both of whose parents carried the mutation. This is because, as mgkrebbs very correctly pointed out in the comments below, otherwise her sons would mate with women who are not their sisters and their children would have the mutation but not her mDNA. If the phenotype is only expressed when the mutation is homozygous, only women whose mother carried the mutation (and Eve's mDNA) would show it and only then would a new mEve arise.