tl;dr: avoid using onmousemove events with Google Chrome
The first step in troubleshooting is to duplicate the problem. In this case, the page worked fine for me in Firefox, so I tried using the same browser as the reporter: Chrome. Sure enough, I could no longer hold down the mouse button and select text on the page. Now that the browser was implicated, it was time to see what it was about this page that caused the problem.
It seemed fairly unlikely that something like this would go unfixed if it was happening on the flagship MediaWiki site, Wikipedia. Sure enough, that site worked fine, I could select the text with no problem. Testing some other random sites showed no problems either. Some googling indicated others had similar problems with Chrome, and gave a bunch of workarounds for selecting the text. However, I wanted a fix, not a workaround.
The extension in question, known simply as "balloons", has actually been removed from the MediaWiki extensions site, for "prolonged security issues with the code." The extension allows creation of very nice looking pop up CSS "balloons" full of text. I'm guessing the concern is because the arguments for the balloons were not sanitized properly. In a public wiki, this would be a problem, but this was for a private intranet, so we were not worried about continuing to use the extension. As a side note, such changes would be caught anyway as this wiki sends an email to a few people on any change, including a full text diff of all the changes.
// Track the cursor every time the mouse moves document.onmousemove = this.setActiveCoordinates;
Sure enough, duck-duck-going through the Internet quickly found a fairly incriminating chromium bug, indicating that onmousemove did not work very well at all. Looking over the balloon extension code, it appeared that onmouseover would probably be good enough to gather the same information and allow the extension to work while not blocking the ability for people to select text. One small replacement of "move" to "over", and everything was back to working as it should!
So in summary, if you cannot select text with the mouse in Chrome (or you see any other odd mouse-related behaviors), suspect a onmousemove issue