I found the answer to this question while writing it, so I've broadened it a little. I wanted to access the --servername argument, in order to create dynamic settings in my .vimrc file.
Through vim's help, I found the v:servername variable, and my script is working. However, now I'm curious if it's possible to access any arbitrary command-line argument. For example, if I wanted to know if vim was in Lisp mode (-l) or Debugging mode (-D), how would I do it? There seems to be no corresponding v: variable for them.
Here are the variables I found by autocompleting :help v:<Tab>

Is there a general way to access command-line arguments from vimscript?