Remove bash startup behaviour (mac user)

If it only happens for one user account, check the following files:

• ~/.bash_profile (and any files sourced therein)
• ~/.bashrc (if it exists)
• your Terminal preferences: Under the Startup tab, is there a command defined to be run when starting Terminal?
• your Terminal preferences again: Under the Settings tab, check to see if your selected terminal style/configuration defines a startup command under the tab Shell.

As per the comments to this post:

• Check /etc/motd to see if a message of the day is defined for login

You can try the following

touch $HOME/.hushlogin

If the message still shows up, it is probably located in one of the bash configuration files. Check those for an echo or print command.