A nipcheese is a penny-pincher or skinflint, all three suggesting a sordidly covetous or penurious person who cuts the cost or quantity of everything to the minimum, often to his own benefit.
Nip here is in the sense you might use when speaking of nipping off the heads of dead flowers or of nipping some enterprise in the bud, stopping it before it gets fully underway. Cheese features in the word because it’s a staple food whose portions can easily be reduced by trimming them, an idea that we also have in cheeseparing.
Nipcheese began life as a seafaring term, a nickname for a ship’s purser, the officer responsible for provisioning and keeping the accounts. Pursers were a notoriously hard-hearted and tight-fisted breed of men, as Francis Grose explained in an entry for nipcheese in his Classical Dictionary of the Vulgar Tongue in 1788: “A nick name for the parser of a ship: from those gentlemen being supposed sometimes to nip, or diminish, the allowance of the seamen, in that and every other article.” They were often suspected of keeping the savings for themselves:
There’s Nipcheese, the purser, by grinding and squeezing,
First plund’ring, then leaving, the ship like a rat.
A Collection of Songs, Selected From the Works of Mr Dibdin, 1796. Charles Dibdin was a famous actor, composer, and writer of the period, whose songs included Tom Bowling.
As you can see from this example, a couple of centuries ago it was a useful name to give a character of miserly mien. The word appears in recent times only in historical novels:
There’s never been anything nip-cheese about my parties, and nor there ever will be!
The Nonesuch, by Georgette Heyer, 1962.