What was the name of Barney and Betty's adopted son?

Barney and Betty's adopted son was Bamm-Bamm Rubble.

Bamm-Bamm first appeared in the fourth season of The Flintstones, in the 1963 episode “Little Bamm-Bamm.” Barney and Betty found the abandoned infant on their doorstep, with his unusual name identified by a note left with him. After a court case, the Rubbles were permitted to adopt him.

His defining joke was extraordinary strength: even as a toddler, Bamm-Bamm could swing a heavy club and shout his name while effortlessly knocking things around. That trait contrasted with Pebbles Flintstone, Fred and Wilma’s daughter, and helped establish the two children as a pair.

A common mix-up is confusing Bamm-Bamm with Pebbles. Pebbles is the Flintstones’ daughter, while Bamm-Bamm is the Rubbles’ adopted son. In later stories, Bamm-Bamm and Pebbles grow up, marry, and have twins named Chip and Roxy.