When a guy wakes up in the middle of the night, with a huge erection. A guy will try to ignore his erection and try to go back to sleep (because when you're tired, it's hard to wank it).
The problem with this is, the more you ignore it, the harder it gets and the longer it keeps you awake. So you're only choice is to just beat it, ejaculate, and go to sleep in your orgasmic state.
I woke up at 3 in the morning last night, and had a late night stick that won't quit, and I was up 'till 4:30 until I decided to just beat it the fuck up.
Slightly old-fashioned term for what is usually today called a "Billy Club" or (UK) an officer's baton, a rodlike piece of wood, usually painted black, worn by police to strike or subdue suspects.
In the USA many registered, private security guards who are not licensed to carry firearms may carry a Night Stick.
Certain phallic implications are almost inevitable.