The part is a Briggs and Stratton 393456 alternator diode assembly. It has 2 wires(1 red, 1 black) on one end and a plug with 2 pins on the other. I looked up the schematics for its installation and found the diode is in the red wire, so the black wire I did not connect, just tucked it into its own sheath. I connected the red wire to the blue tail near the solenoid, and attached a "bullet plug" on the blue wire to the brake controller, and plugged it onto the pin for the red wire of the diode assembly.
I have to assume I got it right, cuz it works perfectly, and it wasn't without the diode installed. I paid around 20 bucks for the part at a local auto parts store. Could have gotten way cheaper online, but didn't want to wait a week for shipping.
There are probably better options out there, but this one was readily available and easy to install.
Aside from the shrink tube, you wouldn't even know it wasn't part of the original harness. I secured it to the existing wire loom with a small zip tie and tucked it under the mower body above the brake controller, out of sight.