Add date before entry title in Twenty Twelve

Base on @Tunji answer.
You can add some conditions to the function to detect home-page, feeds…

function display_extra_title( $title, $id = null ) {

    $date = the_date('', '', '', false);

    if(is_home()){
        $title = get_bloginfo( 'name' );
        return $date . $title;
    }
    else{
        return $date . $title;
    }

}
add_filter( 'the_title', 'display_extra_title', 10, 2 );

In order to properly return the_date and not echoing it, you must set the last parameter to false. See more details here

If you don’t want to display the date in the menu widget… You can surround $date in a span and add css to not display it when it’s the menu, sidebar…

$title="<span class="title_date">".$date.'</span> '.$title;

In the css (adjust and add all what you need:

.sidebar .title_date{ display:none;}