Based On Post is designed to let users and developers make use of their Theme's sidebars by enabling dynamic content that can be changed by post.
Please make sure that you've placed a checkmark next to it under the 'screen options' tab inside the 'Add Post' or 'Add Page' screen. Sometimes wordpress automatically hides extra fields.
To turn the 'Post Title as Widget Title' option on or off, go to the 'Appearance->widgets' section of your wordpress dashboard, and open the 'Based On Post' Widget. There is a single checkbox to turn it on or off. (see Screenshot 1)