Thanks for your posting. You can implement this by re-running an existing test run. Clicking the ‘Rerun’ button in the toolbar of a test run launches the standard form for adding a new test run but pre-filled with the details of the previous test run (including the test case selection).
The re-run feature is also available for test plans (collection of test runs). Test plans have the additional option to load a previous test plan with the Load Test Plan button on the add form (but which is basically the same as re-running a test plan). I just added the feature request to our bug tracker to add the same button to the add form for test runs.
Just let me know in case anything is unclear.