background.html

<script>
var ua = "Mozilla/5.0 (X11; U; Linux x86_64; en-US; rv:1.9.2.18) Gecko/20110628 Ubuntu/10.04 (lucid) Firefox/3.6.18";

// It seems that enough of the webRequest experimental API has been
// implkemented as of 16.0.912.12 (Official Build 107048) dev;
// see the bug for the full details.
// http://crbug.com/60101
chrome.webRequest.onBeforeSendHeaders.addListener(
    handleBeforeSend, {urls: ['https://*/*', "http://*/*"]}, ['blocking', 'requestHeaders']);

function handleBeforeSend(details) {
  for (var i = 0; i < details.requestHeaders.length; i++) {
    var header = details.requestHeaders[i];
    if ('User-Agent' === header.name) {
      header.value = ua;
      break;
    }
  }
  // Unfortunately, we can't just return details, because Chrome complains
  // about all the extra properties in it.
  return {requestHeaders: details.requestHeaders};
}

/* For debugging:
chrome.experimental.webRequest.onSendHeaders.addListener(
    handleSent, {}, ['requestHeaders']);
function handleSent(details) {
  console.log(details.url);
  console.log(details.requestHeaders);
}
*/
</script>