Transliteration and Meaning:
The Greek word Γάϊος (transliterated as Gaios and pronounced GAH-ee-os) is a proper noun used as a name in the New Testament. It is a common Roman name, equivalent to the Latin Gaius. The name Gaios does not have a distinct meaning in Greek, as it is a transliteration of a Latin name, which was very popular during the Roman Empire. The name itself has no explicit meaning in Latin either, serving primarily as a personal name.

Lexical Insights and Biblical Usage:
In the New Testament, Gaios is mentioned in several passages, representing at least three different individuals:

1. Gaius of Corinth: Paul baptized this Gaius, and he is mentioned in 1 Corinthians 1:14, “I thank God that I baptized none of you except Crispus and Gaius.” This Gaius was likely a prominent member of the Corinthian church and possibly hosted the entire congregation in his home (Romans 16:23).
2. Gaius of Macedonia: Another Gaius is mentioned in Acts 19:29 as one of Paul’s traveling companions from Macedonia who was seized by a mob during the uproar in Ephesus.
3. Gaius of Derbe: This Gaius is mentioned in Acts 20:4, where he is listed among those who accompanied Paul to Asia.
4. Gaius, the recipient of 3 John: This Gaius is the recipient of the Third Epistle of John. John commends him for his hospitality and faithfulness in the truth (3 John 1:1).

According to BDAG (Bauer, Danker, Arndt, and Gingrich), the name Gaios reflects the widespread use of Roman names in the early Christian communities, indicating the integration of various cultural backgrounds within the church. Thayer’s Greek Lexicon and other sources confirm the same usage, highlighting that Gaios was a common name and not indicative of any specific characteristic beyond the individuals it names.

Classical Greek Usage:
The name Gaios is primarily found in Roman contexts, given its Latin origin. However, the usage of the name in Greek literature is not prevalent outside the biblical and early Christian writings. In classical Greek literature, names of Latin origin like Gaios would typically be found in historical texts dealing with Roman affairs.

For example:

1. Gaius Julius Caesar is often mentioned in Roman historical accounts, though these are usually in Latin or translated into Greek by later historians.
2. Gaius Octavius, known later as Augustus, is also referenced in historical records, particularly in works that discuss the transition of Rome from Republic to Empire.

These examples show that while Gaios was a common name, its presence in Greek literature is more a reflection of the Roman influence than a native Greek usage.

Summary:
The name Γάϊος (Gaios) appears in the New Testament as a personal name for several individuals, indicating its popularity within the Roman world. The name is used for a few key figures in early Christianity, including a host for Paul in Corinth, a companion on his missionary journeys, and the recipient of a personal letter from the Apostle John. While the name itself does not carry any significant meaning in Greek, its usage in the New Testament highlights the diverse cultural background of early Christians and the integration of Roman names into the Greek-speaking Christian communities. The name Gaios reflects the broader context of Roman influence in the early church, both in terms of individuals and the spread of Christianity within the Roman Empire.