I have read the documentation available at Child Themes, Templates
It promises that "your child theme can overwrite any file in the parent theme: simply include a file of the same name in the child theme directory, and it will overwrite the equivalent file in the parent theme directory."
Parent Theme: TwentyTwelve (v1.1)
Child Theme: TwentyTwelve-child (an original name thunked-up by me)
» Given: if I copy my parent header.php to header.php.BAK, make specific changes to the parent theme's header.php, the changes work on the site when TwentyTwelve is activated.
» Given: I move the modified header.php from my parent into my child, and rename parent's original and untouched header.php.BAK to header.php.
» I activate my child theme, TwentyTwelve-child, which now contains the modified and "it worked when it was a parent" header.php.
» Results: the promise of my child's header.php overwriting (or more precisely, overriding) my parent's header.php fails to be realized. The changes don't occur.
Insert proverbial and oft heard "HELP!" followed by "pretty please?" and a grateful and heart-felt"Thank you!"