How does geoprocessing results window store tasks?

This looks like a bug to me because Using the Results window 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.].

As commented by @user23715:

Bug is 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.

I found a workaround for persistent rename:

  1. Rename a process
  2. Drag to “Shared”
  3. Save mxd

In the Shared container the process retains the tool name, e.g. "Append", but automatically get's a new parent with the wanted label.

Think ahead because once in Shared it can't be renamed again.

Step one, rename the Session result

After close and re-open:

Step 4: after re-open mxd