Drupal - How to Rename "Request new password" link?

The String Overrides module is a good suggestion if you have many strings, or need to constantly alter them, but you can also achieve this by editing the string overrides section of your settings.php file.

Simply uncomment the section so it looks like this:

$conf['locale_custom_strings_en'] = array(
  'Request new password' => 'MY CUSTOM TEXT FOR A NEW PASSWORD',
);

Have you tried the String Overrides module?

Provides a quick and easy way to replace any text on the site.

Features

• Easily replace anything that's passed through t()
• Locale support, allowing you to override strings in any language
• Ability to import/export *.po files, for easy migration from the Locale module
• Note that this is not a replacement to Locale as having thousands of overrides can cause more pain then benefit. Use this only if you need a few easy text changes.

I just tried this in Drupal 7 but it wasn't quite right. It needs an extra [''] bit added. This works:

$conf['locale_custom_strings_en'][''] = array(
  'Request new password' => 'MY CUSTOM TEXT FOR A NEW PASSWORD',
);