The river that thrice forks into two streams and reunites a few miles farther on, thus forming the islands of Srirangappattanam, Sivasamudram, and Srirangam is the Cauvery (also spelled as Kaveri) River. Therefore, the correct option is Option 1: Cauvery.

The Cauvery River is one of the major rivers in southern India, flowing through the states of Karnataka and Tamil Nadu. It is known for its religious and cultural significance, as well as its contribution to agriculture and irrigation in the region.

The river splits into two branches near the island of Srirangam, forming a triangular shape with the main river channel. This division creates a series of islands, including Srirangappattanam and Sivasamudram, before the streams eventually reunite a few miles downstream.

Srirangappattanam is an island town in the Mandya district of Karnataka. It is famous for the historical fort of Srirangapatna, which served as the capital of the Kingdom of Mysore during the reign of Tipu Sultan. The Cauvery River splits into two branches near this island, forming a natural boundary for the town.

Sivasamudram is another island located in the Mandya district of Karnataka. It is well-known for the Sivasamudram Falls, which is one of the most significant waterfalls in India. The river divides into two streams just before the falls, creating a scenic spectacle before the streams merge again downstream.

Srirangam is an island situated in the Tiruchirappalli district of Tamil Nadu. It is renowned for the Sri Ranganathaswamy Temple, one of the largest functioning Hindu temples in the world. The Cauvery River splits into two branches near Srirangam, encircling the island and reuniting a few miles away.

In summary, the Cauvery River is the one that thrice forks into two streams and reunites a few miles farther on, forming the islands of Srirangappattanam, Sivasamudram, and Srirangam. Its unique geography has contributed to the cultural and historical significance of these areas in southern India.