How can I make the crossed-"O" ($\oslash$) bold?

There are several options for obtaining bold characters in math mode. Using the bm-package with \bm is considered the best practice for that.

Implementation

\documentclass[border=3mm]{standalone}
\usepackage{amsmath}
\usepackage{bm}
\begin{document}
Options for bold $\oslash$:
$\boldsymbol{\oslash}$ vs.
{\boldmath $\oslash$} vs.
$\bm{\oslash}$
\end{document}

Output

You can use the bm package to get bold math characters. See

How can I get bold math symbols?

However, for your actual purposes, (linguistics) you should not be using the math symbol \oslash anyway, but you should just use the regular upper case Ø, in which case it can be made boldface without any extra packages.