The story
Possibly a euphemistic ellipsis of “to give a monkey's toss”, “to give a monkey's shit”, "to give a monkey's fart", or “to give a monkey's arse”; compare give a rat's arse or give a rat's ass.
Etymology from Wiktionary, CC BY-SA 4.0.
give a monkey's
Today it means: To care; to have an interest or concern. Full definition →