Until recently, upon death a Facebook member's account was frozen. On occasion, Facebook fought surviving family members in court to block the family members' access to the deceased member's Facebook page.
For example, Facebook successfully quashed a subpoena by the family of a woman who died after falling from the twelfth floor of an apartment building. A coroner determined that the cause of death was suicide. The family disagreed and wanted to access the woman's Facebook page because they believed it would contain evidence of her state of mind indicating that she did not commit suicide. In an opinion available here, a federal court in California sided with Facebook, finding "that civil subpoenas may not compel production of records from providers like Facebook [and that] [t]o rule otherwise would run afoul of the 'specific [privacy] interests that the [federal law] seeks to protect.'"
With the new "legacy contact" policy, Facebook users are given the option of naming a person who will have some ability to manage the account post-death. The policy is optional. If a Facebook user does nothing, the account will be frozen, which Facebook refers to as "memorialization." A legacy contact will have the ability to write post-death posts, change the deceased's profile picture, and respond to new friend requests. The legacy contact will not be able to edit content created by the deceased while he or she was still alive.
Users can select a legacy contact by going to Settings and then Security and then choosing Legacy Contact. In addition, Facebook has stated that it if a member executes a Will giving someone rights to digital assets the company will treat that person as the "legacy contact" even if that designation was not confirmed on the company's website. For more information on rights to digital assets see this post on the Uniform Fiduciary Access to Digital Assets Act.