Three ideas for you:
1) Refugees fleeing from Augustus's rule after Marc Antony's fall in the East settle on Sri Lanka, eventually forming a large city and slowly taking over the island under a Republic that has many of the scrolls from the Great Library. It slowly grows to take over parts of Southern India and colonizes much of modern Malaysia, Indonesia, Australia, and more. By 1000 AD the maritime empire stretches from South Africa to Madagascar to much of Southern India to Australia to New Zealand and Formosa, perhaps Sakhalin Island. They trade with everyone and have become very rich, as well as the undisputed maritime masters of the world.
2) Traders from Rome establish a large enclave in the Southern part of the island that becomes gorged with refugees as Rome falls and again as Islam moves through the Levant and Egypt. They slowly grow to take over the island and have colonies in various areas, but slowly lose their Roman identity as a polyglot Republic emerges. Its technology and martial prowess are undisputed as they must face stiff Tamil resistance on the mainland to expand, eventually focusing on naval and maritime trade running from southern Africa to Egypt to India to China to New Guinea and perhaps to medieval Japan too.
3) Roman refugees merge with a local government and form a quasi-Roman state with very unique architecture and a penchant for conquest, they survive for centuries and have a strong Tamil and Buddhist influence.