In the past Buford won a goldfish at a carnival which he named "Biff" after his mother, Biffany. When a bully tried to eat Biff, Buford became a bully himself in order to get him back. When Biff disappeared, Buford became extremely emotional and sensitive, revealing how much he cares for Biff. He then saves Biff again, this time from being eaten by a giant squid, and apparently loses the fear of squids after saving him ("Voyage to the Bottom of Buford").
In the Phineas and Ferb video game, Buford's fish is named Bubba, but it might be another fish other than the non-mentioned Biff. Then again, it could have also been a mistake.
Biff and Buford could be a reference to "Back to the Future Part III" as Buford "Mad Dog" Tannen is Biff Tannen's ancestor.
Buford stated that he named his fish after his mother.