Your guess is as good as ours.
Sometimes we get a Question of the Day that proves utterly baffling and this is one such instance. As you can see in the accompanying photo, the property was called "Biff's Famous Food and Las Vegas Club." So it wasn’t, as you say, called Biff’s Las Vegas Club. The "Famous" adjective modified the "Food" noun. In fact, in another (unavailable) photo, under Biff’s Famous Food, a smaller sign read, "Specialties Miner’s Chuck [and] French Dip."
However, if "Biff" was a real person, he has left no traces other than an old long-since-removed sign. (The Biff's era, in the mid-’50s, from what we can tell by this and a few other photos, preceded the makeover of the Las Vegas Club into a sports-themed property.)
We consulted several local historians, who came up empty. We asked former Las Vegas Review-Journal publisher Sherman Frederick, an old golf partner of the late Mel Exber, onetime co-owner of the Las Vegas Club, but got nowhere. We were unable to run Mel's son, Brady Exber, to ground.
As for the Las Vegas Club, it’s presently closed and its future is cloudy, as new owners Derek and Greg Stevens brainstorm what to do with the property, having purchased it, the small La Bayou and Mermaids casinos, and the Glitter Gulch strip bar to make way for something new and undoubtedly exciting.
There’s a separate "Biff" angle to Fremont Street: In Back to the Future II, Marty McFly's nemesis, Biff Tannen, owned Biff Tannen’s Pleasure Palace Casino Hotel, whose entrance, tower, and lights bore a striking resemblance to the Plaza Hotel, directly across the street from the Vegas Club. (The Plaza was also owned by Gaughan and Exber for a time, until multinational conglomerate Tamares Group bought them out in 2005.) Thus, whoever named the restaurant at the Las Vegas Club "Biff's" was inadvertently clairvoyant. Either that, or whoever named the Back to the Future character knew a little something about an obscure moment in Las Vegas history.