This seems like an odd problem, but it’s a real one. And it’s an issue you need to deal with, and soon. We’re not just talking about what happens to your Facebook account, but what about all your bank accounts, email accounts, passwords, and more?
Most importantly, someone you care about/who cares about you needs to be able to access your main email account once you’re gone. They’ll need access to at least one of your email accounts (hopefully, your main one) in order to close various accounts, access online accounts, and close things as necessary. Google makes this simple—if you have a Google account, you can specify what you want to have happen in case you haven’t logged into your account for a specified period of time (3 months, for example), through the Inactive Account Manager. If you have a Google account, check out the settings here: https://www.google.com/settings/account/inactive.
For everything else, we strongly recommend (as we have several times in the past) using a password manager, such as 1Password (http://www.agilebits.com/onepassword), LastPass (http://www.lastpass.com), or Dashlane (http://www.dashlane.com). Each of these has at least rudimentary support for sharing passwords with a loved one. Ken has been using Dashlane recently, and it includes support for one or more emergency contacts. You can specify that these contacts receive the ability to log into your accounts should something happen to you. The details vary by product, but it’s certainly worth checking them out.
You could, of course, do what most people do—keep a paper list of all your passwords. In that case, you’ll need to work a little harder to get your account information to your digital executor(s) after your death. Using a tool like a password manager, however, makes things a lot simpler, assuming that you’ve set up your emergency contacts before that big software bus in the sky plows you down.