In a 2015 Interview with the National Post their son Michael said that the Berenstain's ancestors had emigrated to the USA from Ukraine in the 19th century and thought that the -stain spelling, which differs from the more common -stein spelling, was an attempt to phonetically render the way they pronounced their name. In some forms of Yiddish, the -stein ending is pronounced -stain, and it's possible that their ancestors only had documentation written in Cyrillic or Hebrew script to prove their identity to immigration officers and this was how it was transcribed into English.
A popular meme has it that their creation used to be spelled "Berenstein Bears", but it was changed or was that way in a parallel universe.
Here's a scan of the first Dell edition of their book "Marital Blitzed", published in 1955, well before the bears' debut in their 1962 book, "The Big Honey Hunt": Note the spelling, 'Stanley and Janice Berenstain'.
