Pronounced "biffy", BFFI is used as a term of endearment for a great friend, whom you hope to be friends with longer than life, or forever, but rather, infinitely.
My BFFI and I are going to the mall. No, you're not invited, Paul.
Bailey: What about Whitney? You two are friends, right?
Fuentes: Yeah, he's my BFFI, but its just not the same. Ya know?