After the childless Rubbles wish upon a star for a baby of their own, Barney finds a basket containing the world's strongest foundling. However, someone else wants to adopt little Bamm-Bamm, and he's rich.
Fred takes Pebbles to the supermarket and discovers she puts a lot of things in the basket. Fred is afraid his daughter might be a budding kleptomaniac. While Fred's at the jeweler's to purchase a gift for Wilma, a thief plants an expensive diamond collar in Pebbles's crib. Back home, Wilma is delighted with the "gift," but Fred suspects that Pebbles snatched it. He goes back at night with Barney to the jewelry store to sneak the collar back, but the thief is waiting there to recover his loot.
Fred's beginning to think that taking Pebbles for walks in her stroller is bad for his image. Then he hears of a club in which fathers can park their babies' carriages while they go attend to other things.
The Flintstones and the Rubbles are fans of "Peek-a-boo Camera," a show that catches people off their guard as they go about daily life. Fred and Barney are invited to a bachelor party thrown by the Water Buffaloes. The boys lie to their wives that they are going to visit a brother Buffalo in the hospital, then attend the party instead--and are filmed there by the crew of "Peek-a-boo Camera."
Fred finds a cute kitten and brings it home. The little feline has an enormous appetite, and in time it becomes clear that this is no house cat, but a lion. Attached to his pet, Fred avoids facing the reality of what he's brought under his roof as long as he can.
The latest feud between Flintstone and Rubble erupts when Barney votes for Joe Rockhead instead of Fred to be the lodge's next Grand Poobah. Fred tells Barney they're through. Then Barney learns that the new room Fred built extends onto Barney's side of the property line. Now Fred must endure in his own house the presence of his new enemy.
To promote their new movie, "Son of Rockzilla," the producers hire Fred to wear Rockzilla's costume. However, Fred is unable to remove the creature's frightful mask. Now he is being pursued by both cops and an infatuated female Rockzilla-like creature who's broken out of the zoo to follow her new love.