The Chicago Bears got their name from George Halas. He named them based on one of the other Chicago sports team at that time. He got the from the Chicago Cubs, and the reason that he got the name from the Cubs is that a cub is a small bear and since football players are usually bigger than baseball players he thought that the name meant that the Bears would be bigger and stronger. So, the name was appropriate and it stuck.
Here is the history of the Bears logo from 1940 all the way to 1999 when the last change was made to the logo:
Here is the history of the Bears logo from 1940 all the way to 1999 when the last change was made to the logo: