Gómez is a common Spanish surname, used throughout the Spanish-speaking world. It is one of the few Hispanic names that does not come from Latin, arising instead from the Visigothic word guma, meaning 'man'. This is the same root word that gave us the English 'groom' (as in 'bridegroom'), but it has not given us any common English surnames -- the surname Gomme is perhaps the most popular English variant; Gomer, surprisingly, is unrelated, coming from the Greek Γαμὲρ ('Gamér').
The Visigoths settled in Spain in 456 AD, and the name Gamér is recorded as coming out of the Castile region, an area of popular among the nobility. As the Spanish conquered much of the world, including most of South America, the name Gómez followed. While it maintains its traditional form in most of the world, the Portuguese tend to use the form Gomes, and the Catalans Gomis.