<?php
$file = "...\n" .
        "<div id=\"phrase_of_day\">\n" .
        "    <div>\n" .
        "        <span class=\"icon quote\"></span>\n" .
        "        <h1>Frase do Dia</h1>\n" .
        "        <blockquote><p>value to change</p></blockquote>\n" .
        "    </div>\n" .
        "</div>\n" .
        "...";
$o = preg_replace(
  '/(<div id="phrase_of_day">.*?<blockquote><p>)([^<]+)(<\/p><\/blockquote>)/s', 
  '$1hello world$3',
  $file);

print $o;
?>