There are a ton of sites regarding this issue, I spent lots of hours figuring this one out. Every one of those sites forgot to tell you uploading the .HTACCESS file using your FTP client does not work (it did not work for me). I would spend days waiting for the server to update whatever it had to update and nothing happened, the CLEAN URL option remained grayed out no matter what I did.

Then I gave Godaddy's web file manager a chance and IT WORKED, it is like FTP but inside a browser, select the .HTACCESS file and update it with the new code. In my case the .HTACCESS file was empty despite uploading it at least a thousand times with a thousand variations using my FTP client.

I also uncommented the # RewriteBase / line, that means removing the # sign. Also, I read that this line should be right above the following RewriteCond lines. I doubt this actually does anything but I may be wrong.

Anyhow, see instructions below with one-line-modified default code from Drupal 6.12

Please leave your comments below and don't forget to download "AUTOPATH" (http://drupal.org/project/pathauto), it helps you get rid of the horrible NODE in your sites' URL's and customize it to your needs.

INSTRUCTIONS

1. Access your GoDaddy account
2. Click on the Hosting tab and from the drop down choose the first option that reads My Hosting Account
3. Click on Manage Account (new window will open)
4. Click on Your Files (graphic top left)
5. Find and select .htaccess in your file manage on the right and click on Edit (top)
6. Copy and paste the following Apache code (this is the same code from Drupal 6.12 with the uncommented line mentioned above, it is there two times, but the first is the commented one and the second one is the one that lets you choose whether you want CLEAN URL's) into the edit window:

#
# Apache/PHP/Drupal settings:
#

# Protect files and directories from prying eyes.

Order allow,deny

# Don't show directory listings for URLs which map to a directory.
Options -Indexes

# Follow symbolic links in this directory.
Options +FollowSymLinks

# Make Drupal handle any 404 errors.
ErrorDocument 404 /index.php

# Force simple error message for requests for non-existent favicon.ico.

# There is no end quote below, for compatibility with Apache 1.3.
ErrorDocument 404 "The requested file favicon.ico was not found.

# Set the default handler.
DirectoryIndex index.php

# Override PHP settings. More in sites/default/settings.php
# but the following cannot be changed at runtime.

# PHP 4, Apache 1.

php_value magic_quotes_gpc 0
php_value register_globals 0
php_value session.auto_start 0
php_value mbstring.http_input pass
php_value mbstring.http_output pass
php_value mbstring.encoding_translation 0

# PHP 4, Apache 2.

php_value magic_quotes_gpc 0
php_value register_globals 0
php_value session.auto_start 0
php_value mbstring.http_input pass
php_value mbstring.http_output pass
php_value mbstring.encoding_translation 0

# PHP 5, Apache 1 and 2.

php_value magic_quotes_gpc 0
php_value register_globals 0
php_value session.auto_start 0
php_value mbstring.http_input pass
php_value mbstring.http_output pass
php_value mbstring.encoding_translation 0

# Requires mod_expires to be enabled.

# Enable expirations.
ExpiresActive On

# Cache all files for 2 weeks after access (A).
ExpiresDefault A1209600

# Do not cache dynamically generated pages.
ExpiresByType text/html A1

# Various rewrite rules.

RewriteEngine on

# If your site can be accessed both with and without the 'www.' prefix, you
# can use one of the following settings to redirect users to your preferred
# URL, either WITH or WITHOUT the 'www.' prefix. Choose ONLY one option:
#
# To redirect all users to access the site WITH the 'www.' prefix,
# (http://example.com/... will be redirected to http://www.example.com/...)
# adapt and uncomment the following:
# RewriteCond %{HTTP_HOST} ^example\.com$ [NC]
# RewriteRule ^(.*)$ http://www.example.com/$1 [L,R=301]
#
# To redirect all users to access the site WITHOUT the 'www.' prefix,
# (http://www.example.com/... will be redirected to http://example.com/...)
# uncomment and adapt the following:
# RewriteCond %{HTTP_HOST} ^www\.example\.com$ [NC]
# RewriteRule ^(.*)$ http://example.com/$1 [L,R=301]

# Modify the RewriteBase if you are using Drupal in a subdirectory or in a
# VirtualDocumentRoot and the rewrite rules are not working properly.
# For example if your site is at http://example.com/drupal uncomment and
# modify the following line:
# RewriteBase /drupal
#
# If your site is running in a VirtualDocumentRoot at http://example.com/,
# uncomment the following line:
# RewriteBase /

# Rewrite URLs of the form 'x' to the form 'index.php?q=x'.
RewriteBase /
RewriteCond %{REQUEST_FILENAME} !-f
RewriteCond %{REQUEST_FILENAME} !-d
RewriteCond %{REQUEST_URI} !=/favicon.ico
RewriteRule ^(.*)$ index.php?q=$1 [L,QSA]

# $Id: .htaccess,v 1.90.2.3 2008/12/10 20:04:08 goba Exp $

7. Save by clicking on little floppy disk on top left
8. Access your admin account in your Drupal site and you should now enable CLEAN URL's.

OPTIONAL
Download the AUTOPATH module from here: http://drupal.org/project/pathauto
It takes a little bit of time to configure but it is easier than it seems.

Good Luck