How to Add a Friend on Facebook With a Private Profile
Facebook gives you the option to tailor your privacy settings on the site, but regardless of how little information users let you see on their profiles, you can typically still add them as friends. If a user has disabled to ability for others to add her as a friend, try sending her a message encouraging her to request your friendship. If the user also disabled the messaging service, however, there is no way to connect with her as friends unless she decides to seek you out.
Log in to your Facebook account and use the search engine at the top of any Facebook screen to find a friend by name. If you don't see the person you're looking for in the short list of results, click on "See all results" at the bottom of the short list and then the "People" link on the left side of the screen. You can also find people on a mutual friend's profile friend list.
Click once on a Facebook user's name to be directed to his profile.
Click on the "Add as Friend" button directly to the right of the Facebook user's name. A friend request is sent to her account, which she can either accept or decline.
Reach out to a friend who has used privacy settings to disable the "Add as Friend" feature by clicking the "Message" button in the top right corner of his profile. As long as you have not disabled the "Add as Friend" link on your own profile, communicate to the user that you would like him to initiate a friend request for you to approve.