My first car, a used 1963 Chevy Impala was originally known as "Old White" but was eventually known as "Old Rust." Eventually a HS kid bought it to restore it so I hope it is now winning prizes at antique car shows and has a name fitting of its elevated status.
I also had a Dodge Van that kept running and running and running - it became known as "Old Blue." I'm afraid I am not very original!
I have a Toyota Echo, my first car that I drove as I only started driving about five years ago, a little over. I still have it, and don't really have a nickname for it. Not because I think it's silly or anything like that, but just never had one.