This page contains information regarding Star Trek: Discovery, and thus may contain spoilers.
Booker told Michael Burnham that Grudge was so named "because she's heavy, and all mine."
Booker was defensive of Grudge and her name. When Cosmo Traitt threatened to "roast Grudge and feed her to the starving" (adding that she would feed a planet), Book told Traitt to keep Grudge's name out of his mouth and called Grudge "a queen." (DIS: "That Hope Is You, Part 1")
Grudge was created by writer/producer Jenny Lumet. Lumet based Grudge on a Pekingese dog, Nelly, that belonged to her grandmother, Lena Horne. Horne used to keep Nelly in the sleeve of her kimono. According to Lumet, Nelly "never did anything." 
Though Grudge's size is stated to be the result of a thyroid condition, she is played by a Maine Coon, a naturally large breed of cat.