I don’t know which version you’re on, but it has been my experience that you cannot rename a part that is on a serial object. 

It is best to scrap the serial object. You can create a custom field with scrapped serial object info to get history for the newly made serial object.

Hi Aleks. No this is NOT possible in any core version. By the way what is your use case here? There can be other workarounds if you can elaborate the use case more.