This looks like a bug to me because the documentation 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.
As commented 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.] (link here, once it gets published).