Changing The Form

To add the drop down select menu for categories, you simply use the wp_dropdown_categories() function with at least one parameter of show_option_none=Select category. You can also add other options too if you like.

The following code is what I use for the searchform.php here on New2WP.

<!-- The searchform.php template code -->
<form id="searchform" method="get" action="<?php bloginfo('url'); ?>">
    <input type="text" name="s" id="s" size="15" />
    <?php wp_dropdown_categories('show_option_none=Select category'); ?>
		<input name="s" id="s" type="text" class="searchinput" value="<?php if ( is_search() ) echo esc_attr( get_search_query() ); else echo 'Search'; ?>" onfocus="if(this.value==this.defaultValue)this.value='';" onblur="if(this.value=='')this.value=this.defaultValue;" />
    <input type="submit" value="Search" />
</form>

Add a function

In order to hook into the search query that happens when you submit the search form you need to add this to your functions.php file.

// Add to functions.php file.
add_action('pre_get_posts', 'search_by_cat');

function search_by_cat() {
	global $wp_query;

	if (is_search()) {
                $cat = intval( $_GET['cat'] );
                $cat = ( $cat > 0  ) ? $cat : '';
                $wp_query->query_vars['cat'] = $cat;
    }
}

If you have any questions about this please post a comment below and let me know. This is something that is so easy though, that it took me just about 10 minutes to add to New2WP on all pages besides the homepage. It shouldn't be too difficult to figure out, but let me know if you need help I'm happy to help with anything.