Open the HTML on the page in your browser and search for "og:title". That should be set to whatever the current menu item title is. If it's correct, but you're still seeing Home, it just means that Facebook hasn't re-scraped your page since you changed things. You can use the
Facebook URL debug tool
to have Facebook re-fetch the page and grab the new content.
If you want even more control over the title, you can add our EasyTag to the page to set the title, like:
{SCOpenGraph title=Your Best Title}
That allows you to set the menu title to one thing and use a different title for social sharing.
I hope that helps explain, but if you need anything else, just let me know.
Thanks,
Alex