The name of the Berenstain Bears series did not change from "Berenstein" to "Berenstain"; it has always been spelled "Berenstain." The confusion arises because "Berenstain" is an unusual spelling, and many people have misremembered it as the more common-sounding "Berenstein." This has happened since the original creator, Stan Berenstain, was even in elementary school, where a teacher incorrectly told him his name was "Berenstein." The family name "Berenstain" reportedly came from an immigration officer's phonetic spelling of the original family name "Bernstein" when the family immigrated from Ukraine. This led to the unique spelling "Berenstain," which has been used in the books since their inception in the 1960s. The misspelling has become famous enough to be part of a larger cultural phenomenon known as the Mandela Effect, where groups of people insist on an alternate version of history or facts, often accompanied by theories about parallel universes. However, all official records and the family themselves confirm the spelling has always been "Berenstain".
In summary, the Berenstain Bears title did not "change" from Berenstein to Berenstain; it has been consistently spelled Berenstain, and the confusion is due to common misremembering of an uncommon name spelling.