Comes from the way pissed up folk say "best friend." To be a "bessfriend" for any length of time is incredibly unfortunate. You attain this dreaded status by having a drunken person latch onto you and tell you all their problems, cry on your shoulder and generally kill your groove. Several times, through hiccups and sobs about ex partners, you will hear the words "You're like my bessfriend, you are." Note- you don't even have to really KNOW this person to be their bessfriend for the night.
The worst bessfriend situations occur when they are members of the opposite sex. You either get cock blocked or pussy locked all night by the weepy eyed, wretched looking individual clinging desperately to your wrist.
"Why didn't you come to the club last night? We all wondered where you got to. Did you go home?"
"No such luck. Some mental bitch in Yates's decided to make me her "bessfriend" for the night, and I just could NOT shake her!"