I want my users to login through 'Last Name', rather than 'Password'
You can do this by using a module called String overrides to redefine the strings exposed to your users during login and on their profile page.

Navigate to Administration » Configuration » Regional and language  » String overrides and fille in the following and click "Save configuration":



Now, login will look like this to your users:



Note that this just renames the fields.  This means that the last name will not have its own field and it will not be displayed on the user's profile page.  It is not clear from your question whether this will be a problem.

However, if you want a real "Last name" field, and you want to use this field for authentication, you can do this by creating a small module that uses hook_form_FORM_ID_alter() to make a couple of changes to the login form:


Hide the password field.
Add "Last name" as a field.


Now, in the form validation, check that the value of the "Last name" field matches the last name in the user's profile, and if it does, fill in the hidden password field with the correct password.  If you use the same password for all users in this set up, it can be hard-coded in your form validation.  This will make Drupal take care of all the heavy lifting of creating a session, book-keeping of logins, etc.

As for the (in)security of using the last name instead of a real password as a login credential, I'll take your word for security not being an issue here.  However, I am puzzled by why want this, so I have added a question about it as a comment.