In English, what does the Latin word gaius mean?

Happy, rejoice.


http://www.thinkbabynames.com/meaning/1/Gaius

The origin of the name is not entirely clear, so it's difficult to translate.

It could be related to the Greek name for the Earth or mother goddess -- Gaia. Thus "Gaius" could be a man of the earth.

Alternatively, it could be related to the Latin for "rejoice" -- gaudere.

http://babynamesworld.parentsconnect.com/meaning_of_Gaius.html

The only known original Roman etymology of Gaius is expressed as a gaudio parentum, meaning that the name Gaius stems from the Latin verb gaudere ("to rejoice", "to be glad"). This etymology is commonly seen as incorrect, and the origin of Gaius is often stated as still unknown. Some have linked the name to an unknown Etruscan phrase, others to the gentilician name Gavius, which possibly might have lost the medial v in the course of time. But no supporting evidence has been found to this day.

http://en.citizendium.org/wiki/Gaius_Iulius_Caesar_(name)