What could you give if you couldn't give a monkey's? Answer; nothing at all; it means you couldn't care less.
It's a short form of I couldn't give a monkey's toss.
A toss is another word for a throw.
You can toss a pancake - like this.
You can say, I couldn't give a toss.
But best of all, I couldn't give a monkey's.
We can also call children cheeky monkeys - these ones here are certainly naughty and they couldn't give a monkey's about damamging the jeep!