The process of adding individuals to one’s social network; the creation of a social network user’s social graph. 

Friending is the activity of inviting someone to be your friend on a social network, e.g., Facebook. They have the option to accept or reject you. It is the primary activity on social networks which are primarily utilized to “enrich existing relationships.” 

Here’s an interesting perspective on ‘friending’ in the comments to a post on the A VC blog:

 “…Users "friend" each other as a show of affection - its cultural currency on a college campus. As I've seen in my research, users average hundreds of friends - obviously more friends than anyone could juggle. Implicit in this mass "friendship" is that the bond is weak. More or less, people in the Facebook aren't checking all their "friends" profiles every day - nor do they care to.”

For more information on friending, see “Never Ending Friending: A Journey into Social Networking”