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 a big fan of the 'Files for most of its run, but stopped watching around the time Duchovny and Anderson were both taking time off. I've caught some of the Doggett/Reyes episodes in syndication, and surprised myself by actually liking many of them.