I've thought that it would be a very cool experiment to slowly work a population of mice & a population of rats back toward their common ancestors, to the point where they could mate with each other.
There's probably more to it than simply determining the ancestral DNA sequences & engineering them into the rats' genomes. But maybe not.
There have been some limited attempts to bring back some recently extinct species, by breeding close relatives to enhance the desired characteristics. The quagga is one subect of this. the Quagga Breeding Project .
I had a more mundane idea. You can often figure out an organism's environment from the properties of some of its proteins. For example, animals that live at low oxygen levels have hemoglobins with very low binding constants for oxygen. So, if you figured out the sequence of several proteins from the Murinae common ancestor, you might be able to figure out quite a bit about how it lived, even though you had no idea what it looked like.