You need to create a copy of page.php and call it home.php, removing the call for the title, and WP will automatically use that for your site's index page.
Do not delete page.php, and you will need to hard-code in your homepage content into home.php, as this page won't be able to be edited from the normal write/manage->pages section of admin.
Other than that, if you know a little PHP, you can use "if" statements to exclude the title from that particular page. I'm just not good at it, so I use the option I mentioned above. Maybe someone else can come in and give you the PHP code.