You'd need to break permissions inheritance on the Home page in the Site Pages library to include only those who you wish to have access to view it. For those that you do not wish to have access, you're going to need to give them some alternative URL to get into the site.
Typically when someone tries to access http://sp.domain.com/sites/thesite/thesubsite, they get redirected automatically to the home.aspx page. For users that don't have access, they'll get access denied even though they have the ability to access other content. You'll need to create some other landing page for those users and provide them the URL so they can get into the subsite.