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") Book apparently called Grudge a queen frequently enough that Burnham teased him about it. (DIS: "People of Earth")
Grudge, along with Book and his vessel, traveled to Earth with Discovery and remained in the region afterward. (DIS: "People of Earth") About a month later, after Discovery reunited with Starfleet command, Booker's ship arrived unexpectedly with only Grudge aboard. The crew discovered he was exploring a lead on a black box and had set his ship to deliver Grudge to Discovery if he was detained. Burnham left Grudge in her shared quarters with Ensign Sylvia Tilly, who was wary of the animal's size, before departing on Booker's ship on a rescue mission. (DIS: "Scavengers")
When Lieutenant Detmer and Ryn took Booker's ship into battle against the Emerald Chain flagship Viridian, Grudge leapt onto Ryn's lap, much to the Andorian's alarm as he had never see a cat before. (DIS: "The Sanctuary")
Grudge accompanied Booker on his initial exploration of the Verubin Nebula, where they were exposed to significant radiation. After emerging, both were successfully treated with DNA recombination. However, Booker noted that Grudge had been favoring her left front paw, and Dr. Tracy Pollard agreed to examine her further. (DIS: "Su'Kal")
Grudge was aboard Booker's ship when he crash landed it into Discovery's shuttlebay as Osyraa piloted it into the Federation's distortion field. Before he and Burnham left to retake Discovery, Booker noted Grudge was "safe and secure" aboard his ship. (DIS: "There Is A Tide...") Later, as Booker and Zareh fought aboard an open turbolift car, Zareh mockingly asked Booker whether he or "that fat cat of yours" would leave a larger stain from a fall. Booker overcame Zareh and threw him to his death instead, shouting after him, "She's… a queen!" (DIS: "That Hope Is You, Part 2")
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.