(Simple Mechanics of Materials) Is there a sign convention for where x=0 is and x=L is on a simple horizontal axially-loaded rod? It seems that no matter where the fixed end is, the x=0 end is always on the left end of the rod, so is that a defined convention or can we choose either end to be where x=0?
You can choose whatever you want. Or, more importantly, whatever is best to help solve your question.
There is a natural bias to put $x=0$ on the left side simply because, well, we're used to having the positive number line (from 0 to $L$) go to the right. But you could obviously just mirror the structure in any case to better suit your needs, or simply go with the positive number line to the left.