I wish to alter the default title 's' to another thing for that search widget of wordpress. So, essentially I will have to alter the widget template and also the handler that reads the GET input. Are you able to help me finding individuals two place rapidly please? Thanks.

Looked on the internet "get search form wordpress"

obtained from here: http://codex.wordpress.org/Function_Reference/get_search_form

Drop this function inside your theme's functions.php

function my_search_form( $form ) {

        $form = '<form role="search" method="get" id="searchform" action="' . home_url( '/' ) . '" >
        <div><label class="screen-reader-text" for="s">' . __('Search for:') . '</label>
        <input type="text" value="' . get_search_query() . '" name="s" id="s" />
        <input type="submit" id="searchsubmit" value="'. esc_attr__('Search') .'" />

        return $form;

    add_filter( 'get_search_form', 'my_search_form' );