Why did Betty and Barney Rubble adopt Bam Bam?

Because Barney was infertile.
Hey, did you know that their last name in the German version was “Geröllheimer”?

They were trying the “wham bam thank you ma’am” method, but it didn’t work.

So they went for the “bam” instead.

I always kinda figured the Flintstones took place before biological procreation was invented. You know, they just had bird phones and saber tooth tiger lawnmowers, and I presume, a pterodactyl was on stork duty for all baby making needs.

After being told by Fred that they spent too much time playing with Pebbles and to “get your own baby”, Betty and Barney wished upon a star for one. Soon after, they found an abandoned infant on their doorstep, with the name of “Bam-Bam”, according to the note left in the basket. After a court battle in which they defeated noted lawyer, “Perry Masonry”, who was representing a wealthy man who wanted to adopt Bam-Bam, the couple were allowed to adopt Bam-Bam legally and permanently.

I saw that episode (although I don’t remember any legal proceedings in it).