This looks like a bug to me because the documentationUsing the Results window says that RenameRename 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 SnippetCopy As Python Snippet might be an alternative and perhaps better way to record results and re-run them later.
As commented by @MattWilkiecommented by @mattwilkie:
... 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.].
Bug is (link here, once it gets published)still present in ArcMap 10.8.1 and listed as "In Product Plan"
BUG-000082214 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.