That's what SnekTek decided to do in a recent YouTube video, and the results are pretty awesome. With a little hard work and some light drilling you can add an aux input to your Google Home Mini, too.
The first step is to remove the orange rubber base, which requires heating up the bottom of the device so the adhesive keeping it in place loosens. After that, the bottom cover unscrews easily, revealing a few wires, chips and switches to play with.
After a bit of digging, SnekTek decided to splice the new aux port into the cord that pipes music to the Home Mini's built-in speaker. That meant measuring the voltage and adding a 10-to-1 voltage divider. Then they drilled a hole in the base of the device before using some epoxy adhesive to keep the new input in place.
Finally, SnekTek confirmed that this method works by plugging it into another speaker. So if you're willing to take the risk of trying a DIY upgrade on your new device, go for it. However, if you're worried about breaking your Google Home Mini (this definitely voids the warranty), you're probably better off just paying the $59 for a Chromecast Audio.