Not right now and not in its current form
This would be hard to do well in its current form as mentioned in comments and Rand al'Thor's answer. The Area51 statistics paint us a site that is still growing up. Another specific statistic that would hold the challenges concept back is that, as of this post, we only have ~53 avid users. Which is not a lot to carry something like this. Considering average view on main site post is about 30 and meta less. We would not want to introduce this to have it fizzle out and potentially sour people on the idea later.
That leads into a larger issue that affects how this and other whole community ideas. We are a site that brings together so many different genres of crafts and art. We are only scratching the surface right now with things like drawing and crochet for instance but there are many more. To quote Catija in chat about this issue better than I could:
The point I'm trying to make is that "themes" like "Ask questions about owl projects" require two degrees of separation from our scope... Generally, these topic challenges are only one degree away - "Ask questions about Crochet". But, because our topic is so broad, it's unlikely that a large percentage of users will be interested in participating in any one topic... which makes "topic challenges" difficult.
Designing themed based or topic based challenges had the potential to hinder or exclude large groups of users that don't have interest or skill in a particular genre.
For argument sake even if you could get all users interested in drawing you would still have to quantify the challenge somehow. Just a crochet weekly challenge doesn't seem very enticing from the lack of focus.
You could argue that we need to do actual themes like "Doctor Who", "Outlander" or mushrooms. You cannot guarantee that a large portion of the community would show interest.
If you think this is a good idea I would love to see some Challenge theme/question candidates.
This was brought up a couple of months ago in another Meta. If we wanted to do something I think this would be a better start. It would definitely be a community building experience and there would be no pressure to conform to a topic or theme. It most certainly would have an element of question inspiration which the site could use right now.
I would push to get that rolling first but it might have the same issue of not enough users yet to keep momentum.