Like most things, it's pretty simple from the far view. But there are a lot of details, much of which scientists are still trying to figure out.

The short version is, mitochondria, those powerhouses of the cell, have their own DNA, just like some bacteria. They do their own living, dividing, and dying, and have their own lineage. When a cell splits, some of the mitochondria go to each cell. So the mitochondrial DNA will be inherited from whoever donated the cell, or egg in this instance. So the lineage of the mitochondria would be related to the mitochondria of the egg donor and any other offspring she had, but the DNA in the nucleus, the mouse DNA, wouldn't be related to her.

...and that opens the door to mitochondrial diseases, all of which are pretty terrible.