Christopher Groves, 38, and nine-year-old Connor, who lived in Blantyre, North Lanarkshire, died in the incident on the A725 near the Raith Interchange, South Lanarkshire, at around 11.15am on Saturday.
A steady stream of mourners placed floral tributes to the tragic family at the scene, yesterday.
Connor was a pupil at Udston Primary School, in Hamilton, where a special service was due to be held yesterday.
Speaking after the incident, head teacher Elaine Henry paid tribute to a "bright, friendly, happy boy who was valued by all in the school community."
Mr Groves and Connor, an only child, were travelling south from Whistleberry Toll towards East Kilbride when their blue Honda S2000 left the road and went into the river.
Police divers removed the pair from the car but they were pronounced dead at the scene.
The cause of the crash remains unclear, although it is believed no other cars were involved.
Police said inquiries were continuing and they have appealed for witnesses to contact them on 101.