I often ask niche or difficult questions, and as a consequence I get a few Tumbleweed badges.
Very often, these Tumbleweed questions do slowly accumulate views and answers because they're the only quality Google result for the problem, and often they attract answers, many upvotes and favourites (even in some cases gold Famous Question badges) months and years later. They're not bad questions, they're just niche (I'm saying this because I know some notable people have anti-tumbleweed prejudices).
I find the Tumbleweed badge very useful as a notification that my question is getting very low views. I just wrote on meta.superuser.com some of the ways I respond to Tumbleweed badges.
Then I realised I only ever got one badge per site, and I found that this is by design - it's a once-only badge.
I think this doesn't make sense. If the Tumbleweed badge is useful, surely it's useful for what I use it for - a notification that, for whatever reason, the regulars on this site haven't shown any interest in this question, and therefore you should think about whether it can be improved, re-pitched, migrated or promoted, and also, lower your expectations for how quickly it will get answered.
If there's a serious concern about people becoming "tumbleweed addicts" or trying to inflate their badge total by asking deliberately-boring or needlessly-niche questions, a compromise option would be to add the new question to the existing Tumbleweed badge, and notify the user the usual way. So, if I had 3 Tumbleweed questions, I'd only have one badge, but it would list all three questions and I'd have been notified each time.
I'm not bothered about the number of actual badges I get, I'd just like to be notified when a question goes tumbleweed on me.
Maybe also, it'd be interesting to be able to browse my tumbleweed questions (not just the first from each site) and see what became of them. I know at least one of my questions that began as tumbleweed is now at +7 with 29,815 views, and several others got great answers and thousands of views eventually. It'd be interesting to see how others fared over time.
Even better still, would be to be able to browse all tumbleweed questions on a site, looking for good but difficult or niche questions where a good answer would really make someone's day. Special rewards for answering tumbleweed questions have been proposed in the past (e.g. the above link and this badge suggestion) - I don't think a special reward is necessary, but it'd be nice to be able to tidy up tumbleweed just for its own sake.