I have an outlet adapter that is clicking at 260bpm. What is happening?
This is purely an adapter, i.e. there should be no components in there that go "click". 

This points to the adapter sparking internally, or at the plug. That should not happen. Return the device. You link to a product page of the adapter that's only available through third-party seller on amazon, and these are known to sometimes ship counterfeits. I wouldn't know how to make an adapter cheaper by decreasing conductor distances, but maybe these were factory rejects? (Pretty sure that GE tests everything they put on the market for arcing)

The 260 bpm are probably a measurement error; 60 Hz is 3600 bpm, and that's probably much faster than that app expects, and thus, you get a bogus beat estimate. Try with something less music-centric instead, e.g. the Phyphox app, which comes with audio spectrum plots.

You commented later on:


  @replete when nothing is plugged in, it still makes the noise...when I unplug it, the clicking speeds up, then stops 


Um, what? This is really supposed to be a passive device; there should be nothing inside to produce clicking after completely disconnecting it (I assume you have nothing plugged into the in-, and power output side). 

This invalidates my answer: there's something about this device that is more than just an adapter. (PLEASEPLEASEPLEASE don't be something obnoxious.)