I don't think he is called that for any particular reason. His fellow politicians viewed him as a goofy dumb Cowboy so they called him "Bingo Bob." If he did earn that nickname in some fashion the writers have yet to reveal it.
The episode where he is first mentioned only make reference to the fact that he is perceived to be in the pocket of mining interests and is not particularly swift on the uptake. But no actual source of the "Bingo Bob" moniker.
I was actually upset that all three judges didn't know that the first song was "All Mixed Up" by 311. Are you kidding me? I only really liked Blake, Chris S (for his personality) and to a lesser extent Sundance. All the rest can go.