I like you. I like you not. I like you again. Whatever the causes of your fallout and reconciliation may be, re-friending someone on Facebook is almost as easy as unfriending him.
Re-Adding a Friend
Type your deleted friend's name in the Facebook search bar, or locate your friend in either a post a mutual friend shared or a comment your old friend made on a mutual friend's post. Click the person's name anywhere you might see it to open his Profile page.
If you can't find your old friend, try to recall if you blocked him. If you did block him, you'll have to unblock him first before you can see him on Facebook.
Click the Add Friend button located on his cover image on his Timeline. The Friend Request is sent to him immediately. If want to explain your change of heart or apologize for something., click the Message button to send him a private message.
Unblocking a Frenemy
Click the small Arrow in the upper right corner of any Facebook page. Select Settings from the drop-down menu.
Click the Blocking option in the Settings page. This is where you can block and unblock people, pages, apps and those annoying game requests. Note that there is also a Restricted List on this page, so that you can limit access to some of your posts to anyone who isn't quite a friend, but not an enemy either.
Click the Unblock link beside your friend's name. Now that he's unblocked, you can send him a Friend Request.
If you can't see your deleted friend on Facebook and you didn't block him, there's a good chance he blocked you. In this case, you'll need to find another way to contact him, such as by asking a mutual friend to intervene on your behalf or by contacting him with an email or -- if you're friends outside of Facebook -- with a phone call.