Facebook gives you multiple options for avoiding unwanted chat requests when you're on the site. You can go offline for any specific contact and remain online for all other contacts. You can also go offline for an entire list of contacts. To make yourself invisible to all your friends on Facebook chat, you must turn off chat altogether. When you are offline, messages from your friends are automatically sent to your Inbox folder. You can view the messages and reply when you go online.

Change Offline Settings

Log in to your Facebook account and locate the Chat pane. Click the gear icon below the list of friends and choose "Advanced Settings" from the menu. If you want to be invisible to a few friends or lists of friends, select the "Turn on chat for all friends except" option and then type the names or lists into the field. To be invisible to everyone except a few friends or lists of friends, select the "Turn on chat for only some friends" option and then type the names or lists into the field. To go offline for everyone, select the "Turn off chat" option. Click "Save" to apply the settings.