Perhaps the name Gaius is derived from Gaia, as Gaius may be a masculine form of that name. Just a theory, probably not right.
The pronunciation rules are contradictory: Gā-ǐ-us [ˈɡaː.ɪ.ʊs]
The English rules result in something like "GAY ih us" (in American English). The Latin IPA rules result in something like "GAW ee us".