This snippet turns - [X] into ☑ and - [ ] into ☐.

(defun sacha/org-html-checkbox (checkbox)
  "Format CHECKBOX into HTML."
  (case checkbox (on "<span class=\"check\">&#x2611;</span>") ; checkbox (checked)
        (off "<span class=\"checkbox\">&#x2610;</span>")
        (trans "<code>[-]</code>")
        (t "")))
(defadvice org-html-checkbox (around sacha activate)
  (setq ad-return-value (sacha/org-html-checkbox (ad-get-arg 0))))

To find this code, I searched ox-html.el for [. Eventually I found org-html-checkbox, which is directly called by org-html-format-list-item instead of being a function variable that you can change. So that meant I needed to override the behaviour of org-html-checkbox through defadvice. You can see above how I wrap advice around org-html-checkbox and replace the return value with my own function. For more about advice, read the Emacs Lisp Intro manual.

To find the hex codes for the UTF-8 characters, I searched Google for UTF-8 checkbox and found BALLOT BOX WITH CHECK. I used the hex code so that I didn't have to worry about encoding issues. I tested it by updating one of my weekly reviews. Tada!

Inspired by Grant from Wisdom and Wonder.