I like the idea. When entering a long post with many formulas, the constant re-rendering of MathJax can be a drag on the browser; in such cases it is preferable to type formulas first and add $ $
later. Having a button/shortcut for this would be handy. Ideally, it would loop between text
, $text$
, and $$text$$
, like the current Ctrl+I shortcut loops between text
and *text*
.
Drawing in an extra button takes a designer's effort but is not a big obstacle: button images differ between sites already. For example:
Math.SE editor buttons
Travel.SE editor buttons 
The bigger deal is the script that handles the button clicks and keyboard shortcuts, along with all Markdown processing. It currently stands at $2459$ lines of JavaScript code named wmd.js
(not WMD). Officially called Pagedown, it is being developed by SE as an open-source project. At some point of its history, it was reverse-engineered from seemingly abandoned obfuscated code (the original author could never be contacted again). See:
The editor is the same on all SE sites; maintaining separate versions would be rather painful, I imagine. But there is no need for that; the script can determine if the site has MathJax enabled, and activate the button and shortcut in this case only:
if (StackExchange.mathjaxEditing) { enable $ button and shortcut }
Done! :) And it's now $2460$ lines of code.