How can I insert some extra validation into the theme options’ validation function using add_filter?

You want to manipulate and return the same input you have coming into the function. In your case that is $input. That is how filters work.

function add_validation( $input ) {
    // here I want to add some validation for a simple field named 'example'
    if ( isset( $input['example'] ) && !empty( $input['example'] ) )
        $input['example'] = wp_filter_nohtml_kses( $input['example'] );
    return $input;
}
add_filter( 'theme_options_validate', 'add_validation' );

If you return something other than what you start with you will be overwriting everything that came before your filter. You can do if you want, just be aware that it is likely to break things.