The archive icon in the sidebar footer flips bb's Thread list preference threadLifecycles between ["active"] and ["active","archived"]. It is the same setting as Organize → Show in the list, or:
bb thread-list prefs set threadLifecycles '["active","archived"]'
bb thread-list prefs set threadLifecycles '["active"]'
The plugin has no server and stores nothing. It calls the built-in thread-list plugin's RPC, so that plugin must be enabled; otherwise the button shows an error toast.