What is the scale length (from nut to bridge) of your banjo, and what note is your bass string tuned to?
Have you tried sipping on a slightly higher bridge?
Is this a fretted or fretless banjo?
It's a fretless. Scale is 26.25 inches per builder recommendation, though ive tried moving the bridge around to no avail. I'm using a 3/4 minstrel bridge, a 1 inch bridge helps barely just a little, a roughly 1.25 inch (!!!) bridge stops the clicking sound but is way too high to comfortably play. I'm tuning the wound bass string down to G or A depending on tuning.
What happens when you slide the bridge an inch and a half closer to the tailpiece, and then retune again? (lengthening the scale, thus tightening the string tension)
It helps a little, but still there, and also makes my high string want to buzz
Did you retune it all again after moving the bridge down?
Also, if the bridge or nut slots were made for steel strings, could be that the fat nylon strings are not seating well in their slots...leading to buzzing.
Tell us more about this banjo... a photo would help.
I feel like most of the fretless banjos I have do this to some extent with the wound string. Nothing too loud and crazy tho. In my case it's usually related to how the skin is interacting with the moisture/dryness in the air. Tighter skin brings the bridge up a smidge. Once there's moisture present its all slaps and buzzes. I generally play loud and have grown to ignore it but i go thru phases where I notice it every once in a while.
What is the configuration of the tuner on your thumb string? I have had instances where I was getting some buzz when it was interfering with the string on that peg.