You're right: it's a stupid badge.
In general the SE Team seems to have more of a sense of humor -- or a different one, at least -- than most of the core participants on math.SE. They are not above putting in certain features that are "funny", or playing April Fool's Day pranks, and so forth.
It's not my cup of tea, but if you don't worry about it, it can't hurt you.
Added: It has been brought to my attention that by accessing the tumbleweed badge you can find a list of the questions for which the badge has been awarded. I didn't know about that, and I agree that it's useful. However, it would be just as useful or more if this list were made independent of the tumbleweed badge and the route to the list made more prominent.
In fact, if you look at the answers to the meta.SO question linked to in Martin's answer, you can see from cletus's answer that originally the tumbleweed badge did not come with this feature. Thus it was originally thoroughly pointless, and the SE team seems to have realized this and reacted to it.
That change was a positive one, but let me also say that I am not thrilled with how often the SE platform changes in subtle (and occasionally less than subtle) ways. This is almost never accompanied by any direct explanation at the level of math.SE: rather, one is apparently expected to frequent meta.SO, various blogs and so forth to stay informed about platform changes. I found this annoying, so I am not as informed on site mechanics as I once was.