Facebook has added an option for users to delegate management of their account for when they die

Facebook announced it will let users designate a Legacy Contact : someone who, after a user's death, can post messages to the user's timeline, respond to friend requests and update the account's profile picture and cover photo. Facebookers can opt to let the legacy contact download an archive of photos and posts shared, but the person won't be able log in as the deceased user or access any private messages.

Facebook users can also tell the company if they want their accounts permanently deleted after they die
If you are interested go to your Facebook page's settings, click Security as shown in photo below and at the bottom of the page you'll see an option to add your new feature legacy contact.

Would you let your Facebook profile live on without you?