If you are lucky enough to own a pet bear, you could give him or her a regular old “people” name such as Margaret or Penelope or Kate or Bailey or Riley.
If your pet bear is extra large, you could name him Godzilla or Bruin or Big Bob or Bart – after the famous and huge Movie Bear – https://en.wikipedia.org/wiki/Bart_the_Bear.
If you have twin bears, you could name them Jack and Jill or Gus and Sam. Look at these adorable bear brothers!
You could name your pet bear after a famous fictional bear such as:
Papa Bear, Mama Bear, or Junior Bear
Pooh Bear from Winnie the Pooh
Baloo from the Jungle Book
Yogi Bear from the cartoon series
Super Ted from the movie
Little Bear from the famous children’s books by Else Holmelund Minarik and illustrated by Maurice Sendak
Rupert – from the vintage children’s book.
Plain old “Teddy Bear” is a great name too.