Good Hank "G.H." Hill (born September 26, 1999; Henry Jeffrey Hill) is the infant son of Cotton and Didi Hill. He first appeared in "Peggy Hill: The Decline and Fall," but was introduced through Didi's pregnancy in "Next of Shin." G.H. bears an uncanny resemblance to his older half-nephew Bobby.
G.H. was born when Bobby was visiting Cotton and Didi. Didi went into labor while Cotton was at a strip club, forcing a then-12-year-old Bobby to drive Didi to the hospital himself and assist with the birth. At the same time, Peggy was severely injured while skydiving, and Didi came down with postpartum depression. Peggy's injuries and Didi's depression led to deep tension in the Hill family, leaving no one truly willing or capable of taking care of G.H., except for Bobby. Eventually, however, the family issues were resolved. Although completely trapped in a body cast, Peggy managed to deal with the depression of her accident by rocking G.H. to sleep in his carseat using only her big toe. Peggy's effort seemed to influence the others to put aside their own problems for G.H.'s sake.
Due to being a baby during the run of the show, G.H.'s importance is more measurable in how he affects others than in his own actions. Cotton is perhaps the one who has grown the most because of G.H. In "When Cotton Comes Marching Home," G.H. makes Cotton contemplate suicide when he's unable to provide for his family. Later, however, Cotton seems to realize that G.H. needs him alive, and decides to bond with G.H. instead (by having the baby play with the gun he was planning to shoot himself with). Nevertheless, Cotton eventually dies ("Death Picks Cotton") while G.H. is still a baby. G.H. was last alluded to in "Serves Me Right for Giving General George S. Patton the Bathroom Key" when Didi brought Hank some of Cotton's stuff, but quickly drove off again. She now lives with another man closer to her own age. Despite the insinuation as a result of Didi's appearance, G.H. did not make an actual appearance in this episode.
After Didi couldn't bare to think of a name him, Cotton wanted to name his newborn son Hank, which was the name he wanted for any son he desired. However, he already had a grown son named Hank. Rather than use a more conventional means such as "Hank Rutherford Hill II", Cotton decided (almost out of spite) to pick the name G.H., short for "Good Hank" because of the fact that he didn't like his first-born Hank and considered him a sissy. Upon G.H.'s arrival, Cotton once mockingly referred to Hank as "B.H." ("Bad Hank"), to Hank's displeasure.