This looks like a bug to me because [Using the Results window][1] says that Rename on the Result shortcut menu ...

> Renames the result. The default name is the tool name followed by the
> time and date.

I would recommend reporting it but in the meantime I would suggest that Copy As Python Snippet might be an alternative and perhaps better way to record results and re-run them later.

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As [commented by @mattwilkie][2]:

> ... the bug is still present in 10.3. I logged it with Esri,
> *BUG-000082214 - [Renaming a result in the Results window is not maintained when an MXD is closed and reopened.]*.

As [commented by @user23715][3]:

> Bug is **still** present in ArcMap 10.8.1 and listed as "In Product Plan"

[BUG-000082214][4] being in the Product Plan suggests that it has been assigned to a programmer to work on for the next release.  If that work is successful and passes Beta testing then I would expect that it will appear in the next release.


  [1]: http://resources.arcgis.com/en/help/main/10.1/index.html#//002100000013000000
  [2]: https://gis.stackexchange.com/questions/48943/how-does-the-geoprocessing-results-window-store-the-tasks/48945?noredirect=1#comment202845_48945
  [3]: https://gis.stackexchange.com/questions/48943/how-does-the-geoprocessing-results-window-store-the-tasks/48945?noredirect=1#comment635888_48945
  [4]: https://support.esri.com/en/bugs/nimbus/QlVHLTAwMDA4MjIxNA==