Your home page has the word home in 3 places, the menu, the breadcrumbs, and in the content. I'm guessing it's the one in the content you want to remove.
You can make a front-page.php template and set that for your home page. Juggledad suggested I do that and it worked great for me. Copy page.php or whatever else that's closest to what you want. You can remove the title from it. Take out:
HTML Code:
<?php the_title(); ?>
and its associated formatting and link code, which is probably everything from <h1> to </h1>, inclusive. As juggledad said, experiment until you get what you want.