I am going to break this into three sections: Thomas in the Old World, Thomas in the New World, Why?
Thomas in the Old World
In the Scriptures, the Apostle Thomas is mentioned only in several places, and nothing specific is recorded after the famous “Doubting Thomas” scene.
However, the Christian communities East of the Roman Empire often attributed their evangelization to St Thomas. In particular, the Christians of South India claimed that St Thomas had travelled to India and converted people to Christianity (Jenkins 2008). Even today the assorted Syriac churches of India are referred to as “St Thomas Christians”, and most of St Thomas’s relics are scattered in assorted shrines in India.
Although today divided into numerous separate church communities, the St Thomas Christians were originally a large part of the Church of the East whose base is in Syria and Iraq. This Church at times stretched from the Eastern Mediterranean into China, although usually thin on the ground in any particular location. Thus, St Thomas became associated with the East in general and with missionary work in far-flung, largely non-Christian lands.
St Thomas was also especially revered in the Greater Syria region, where a variety of apocryphal stories about him circulated (Landau p. 22). One example is The Revelation of the Magi, which based on linguistic and textual analysis is from the 300s or earlier. In this text, the Magi use their own skill and rituals to come into direct contact with the Star of Bethlehem, which descends in the form of a small human figure clothed with light (Landau p. 53). A lot of events happen, but skip ahead to when the Magi return to their land. In this story, St Thomas appears to them many years later to baptize them, celebrate the Eucharistic liturgy, and commission them to evangelize (Landau p. 82 - 88).
The main takeaway here is that St Thomas is already associated with evangelizing distant lands and with assorted miraculous events.
Thomas in the New World
St Thomas actually shows up a bit in early modern Latin America. Let’s take a look at a few examples.
Who: Diego Duran, Spanish Dominican missionary Friar.
Where: New Spain (Mexico).
When: 1500s
What: Diego Duran argued that as God loved all of humanity, he must have sent the Apostles to preach to all nations. Therefore, St Thomas was Quetzalcoatl - both were holy persons, celibate, artistans, fasted, and had disciples (Baldwin 120).
Who: Fernando de Alva Ixtlilxochitl, historian and Nahua-Spanish aristocrat (Baldwin p. 124)
Where: New Spain (Mexico)
When: 1600s
What: In one of his many many historical relations, Ixtlilxochitl posited Quetzalcoatl’s activity among the Olmecs “a few years after the Incarnation of our Lord Jesus Christ” (Ixtlilxochitl p. 21). This could be interpreted as a hint to Diego Duran’s earlier assertion about St Thomas (Cervantes p. 76).
Who: Santa Cruz Pachacuti Yamqui, Andean historian.
Where: Peru
When: 1631
What: In “Relación de las Antigüedades deste reyno de Pirú”, the author asserted that St Thomas had evangelized the Andes and prepared the way for their fuller incorporation in the Church in the 16th century (Andrien 120).
Who: Guaman Poma de Ayala, Andean Historia
Where: Peru
When: 1615
What: In Nueva Corónica y Buen Gobierno, Guaman Poma claimed that St Bartholomew appeared and preached to the Inka and drew a picture of this.
I think there are examples from Paraguay as well, but from the mid-1500s onwards legends of Apostles visiting the Pre-Columbian Americas emerge from multiple sources and locales.
Why?
These myths regarding the Apostles preaching in the New World fulfilled two purposes at the time:
Sources
Alva Ixtlilxochitl, Fernando de. Obras Historicas, Vol 1.
Andrien, Kenneth. Andean Worlds: Indigenous History, Culture, and Consciousness under Spanish Rule 1532 - 1825, 2001.
Baldwin, Neil. Legends of the Plumed Serpent, 1998.
Cervantes, Ferndando. The Devil in the New World: The Impact of Diabolism in New Spain, 1994.
Jenkins, Phillip. The Lost History of Christianity, 2008.
Landau, Brent. Revelation of the Magi, 2010.
Poma de Ayala, Guman. Nueva Corónica y Buen Gobierno, 1615.