My cats have been sitting on my keyboard. Now my HD icon on the top right shows no name. I also had a problem with the name on my back-up, external drive. I'll worry about that on later.
My Imac is working fine but I am a little concerned.
You can change it to have a name by right clicking on the HDD icon on the Desktop and selecting GET INFO and change it there under the Name section to whatever you want.
I don't know what you mean with the external back up HDD though.
On the external, used only for Time Machine, when I went to try Time Machine, it froze my system so I had to disconnect it. I figure I will have to reformat that one.