Edit the following two files accordingly to increase the search box size in Joomla 1.5.

components/com_search/views/search/tmpl/default_form.php:

<input type="text" name="searchword" id="search_searchword" size="30" maxlength="100" value="" class="inputbox" />

administrator/components/com_search/helpers/search.php

// limit searchword to 20 characters
if ( JString::strlen( $searchword ) > 100 ) {
        $searchword     = JString::substr( $searchword, 0, 99 );
        $restriction    = true;
}