Gaius sits next to 5 phonetic neighbors — Caius, Cai, and Gale are the closest matches by sound. Most share Gaius's Latin-origin cluster, so the rhythm and vowel shape carry across the list.
You searched for names like
"to rejoice, happy, glad"
"of the valley"
"beloved"
"from the city of Troy"
"little wolf"
"variant of Kylo; possibly from Greek kylos"
"Yahweh is righteous"
"nobleman of the second rank"
"God supports, God heals"
"friend, companion"
"healer, to heal"
"prince, king"
"from Grantham"
"long-lived, permanent"
"son of Hendrik"
"son of Brice"
"bower, canopy of branches"
"safe haven, shelter, refuge"
"most beautiful"
"the color blue, sky"
"rocky settlement"
Similar sound: These names share phonetic patterns with Gaius — similar vowel sounds, consonant clusters, or rhythm.
Shared roots: Several names on this list have Latin origins, giving them the same cultural heritage as Gaius.
Rhythm: Gaius is a 2-syllable name. Names with the same syllable count tend to feel interchangeable when you're narrowing your list.