A trace of something is just a hint or suggestion of it, a very small amount left behind — like the sad cookie crumbs at the bottom of an empty cookie jar.
There are more meanings we can trace to this word trace. As a verb, trace means to follow the course or trail of something, sometimes even chasing or hunting it. It also means to draw or sketch over a picture on translucent "tracing paper." So you could say that Sherlock Holmes traced criminals throughout London, following even the smallest traces of evidence and tracing copies of the villain's notes for his files.