[NoNeck]

That is an electrical relay commonly used when aftermarket cruise controls are added to a vehicle. Of course some PO might have wired it to something else such as electric fans, etc. When wired correctly, this relay functions to open (set cruise) or close the circuit when the brakes are applied.

Here is a simple test to determine what it’s connected to:
Install the relay.
With a test light, probe the connection points on the left side of the relay. These (2) points are the relay contacts. (as shown on the photo)
One point will be hot (12 volts) and the other not-hot.
Operate stuff in the car until the not-hot goes hot.
If that doesn’t work, turn on the ignition and repeat the test.

It sounds like you have a current draw at that relay when the key is 'on'. The PO also could have wired whatever this operates into the 'brake/hazard light' fuse on the fuse block. I would try to disconnect it altogether and see if it helps your situation?

POP