If you open your example file above and do C-h v TeX-complete-list RET, you see in the value a list item like this:
(LaTeX--after-math-macro-prefix-p 1
(lambda nil
(append
(mapcar #'cadr LaTeX-math-list)
(mapcar #'cadr LaTeX-math-default)))
(if TeX-insert-braces "{}"))
This means that in math expressions, the completion function looks in the variables LaTeX-math-list
and LaTeX-math-default
for completion candidates. LaTeX-math-list
is the one for users to customize. As a test, you can add a comment like this to your document:
% (setq LaTeX-math-list '((nil "rndvar" nil nil)))
put the cursor after the last parenthesis and hit C-x C-e. Now try TeX-complete-symbol
again. For me, it works and looks like this:

Now you have make this addition permanent. You can use the custom
interface, make an addition on a file-local basis:
%%% Local Variables:
%%% mode: latex
%%% TeX-master: t
%%% eval: (add-to-list 'LaTeX-math-list '(nil "rndvar" nil nil))
%%% End:
or even write your own AUCTeX style (would be my choice).
While running the completion, Emacs will probably throw a message like this in the Minibuffer:
TeX-complete-symbol: Wrong type argument: stringp, (if TeX-insert-braces "{}")
This seems to be a bug. Please report it to bug-auctex@gnu.org
.
$\rnd
?