Please see the following related answers:
What is a Mitochondrial Eve and Y-chromosomal Adam
Does mitochondrial eve have to exist?
Is it tautological that all living humans descended from a single male and single female human ancestor?
Because mitochondrial lineages that provide a selection advantage are more likely to spread in a population, Mitochondrial Eve (ME) is more likely to have belonged to such a lineage. However, there is no real reason to suspect that ME in particular, or her descendants, had any special evolutionary advantage over other humans existing at the same time.
The identification of Mitochondrial Eve—the most recent mitochondrial ancestor of all living humans—changes over time as mitochondrial lineages in a population go extinct. When this happens, the assignment of ME jumps ahead in time to the most recent ancestor of the remaining lineages.
Extinction of a mitochondrial lineage can be entirely random (though selection helps), and is a consequence of reproduction in a finite population. The likelihood of lineage extinction is negatively correlated with it's proportion in the population. In a population without selection, each member wouldn't have an equal number of descendants, they would an equal likelihood of having some number of descendants (e.g., their number of offspring would be drawn from the same distribution). Under this scenario, and with a finite population, eventually all lineages except one will go extinct.*
All we know about any particular ME is that she was female, and had at least two daughters who themselves each had at least one daughter.
This simply follows from the definition of ME.
We have good evidence that the current ME lived about 150,000 years ago, meaning we can also be confident that she was an anatomically modern human.
Other than those facts, the current ME is anonymous, and we know little about the total population of human mitochondria at the time she was alive.
While significant beneficial mitochondrial mutations occurred sometime before ME, and some may have appeared for a subset of existing mitochondria, it's unlikely that ME herself was the origin of any functionally important mutations.
So, beyond the general statements that mutation and selection occur in mitochondria, and that ME was a member of the successful lineage who had some daughters, there's nothing particularly special or unusual about her.
*The lineage that remains is said to be "fixed" in the population; the chance of this is equal to that lineage's starting proportion of the population. Note that other mutations can occur in that lineage during the process of it becoming fixed, so that all offspring are part of that lineage, but don't have identical DNA.
