The phrase "Jesus H. Christ" undoubtedly springs from the monogram "IHS" depicted on vestments, and other eccesiastical paraphernalia, in both Catholic and Protestant churches.. Some people attribute it to English phrases ("In his service") but it predates English by some centuries.
"IHS" might be Latin or it might be Greek. Both theories have their proponents.
The Catholic Encyclopedia1 favors the Latin: Iesus Hominem Salvator (Jesus, Savior of Mankind).
Another Latin expression with the initials IHS is "In Hoc Signo (Vinces)" (By this Sign (Conquer)) is the motto adopted by the Roman Emperor Constantine for his legions when he converted to Christianity in 312 CE. According to Eusebius2, prior to the battle with Maxentius near the Milvian Bridge at Rome, Constantine dreamed of a cross of light in the noonday sky and the words "In hoc signo vinces." Constantine had the cross (or Greek letters Chi and Rho) inscribed on his soldiers' shields. Constantine won the battle and converted the Empire to Christianity. The Jesuits (Society of Jesus) probably had this story in mind when they incorporated an "IHS" inside a sun for their insignia.
Another possibility is that the initials are a Latin transcription of Greek: Iesous Huios Soter (Jesus, Son, Savior). This seems implausible because the Greek letter "H" is a Capital "Eta" (a vowel: long "e"). The aspiration, the sound of "H", at the beginning of "huios", is indicated by a diacritical mark over a vowel, looking something like this: 'u. On the other hand, a similar Greek phrase was used by the early Christians: Iesous Christos Theos 'Uios Soter, the initials of which spell Ιχθυς (Ichthus) the Greek word for "fish", which became a widely used secret symbol for Jesus.
Some try to trace the IHS to pagan roots. Christian cranks maintain it refers to Egyptian gods (Isis, Horus and Seth or Seb). This is the theory espoused by Jack Chick in his tract, "The Death Cookie",3 which attempted to demonstrate that the Roman Catholic Church is a Satanic cult.
More likely, interpreting the H as an Eta: "Ies" is reputedly4 one of the many names of the god Dionysus, whose cult shares some features with the Christian (a dying god, sacramental consumption of wine).
2 Eusebius, Vita Const. 1. 28-29.
4 See, e.g. http://www.truthbeknown.com/solar.htm