For a long time, I've admired Kapitza's Problems in Physics [pdf]. It is a collection of 224 questions Kapitza and colleagues used to form the focus of a postdoc candidate oral exams. They really are a treat. I encourage everyone to read Kapitza's introductions of the problems in the first few pages of the linked pdf, but I'll quote some of the relevant parts here:
In compiling these problems I had a certain objective in mind, which explains their unconventional formulation. The following explanation will make their solution of greater interest to the reader.
Problem solving is of particular importance in the study of the exact sciences such as mathematics, mechanics, physics, etc. It enables the student to apply his own knowledge to the solution of practical questions. [...] It is well known that fruitful scientific work requires not only knowledge and understanding but also a capacity for independent analytical and creative thinking. In effect, these problems were compiled as a useful means for the discovery, evaluation and cultivation of these qualities during the teaching process.
I strove to achieve this end by formulating the majority of the questions in the following manner. A small problem is presented, and the student, using the known laws of physics, must analyse and describe quantitatively the natural phenomenon involved. These natural phenomena were selected in terms of their scientific or practical interest within the scope of the students' level of knowledge. [...]
Most of the problems allow a number of approaches to their solution in order to reveal the students' individuality. [...]
A characteristic feature of the problems is that they have no definite answer because the student is allowed to proceed further and further with the analysis of the problem posed, depending on his own abilities and inclinations.
In particular, here are some of my favorite examples.
1) Astronomical observations show that the planet Venus is entirely covered in cloud, so that the Venusians are unable to observe the heavenly bodes, Describe how they could accurately measure the length of their day.
11) Explain why, for a bow of a given size, there is a certain size of arrow which yields the longest flight. Estimate this size for a bow of a given shape.
19) Why can the movement of a bicycle be controlled in 'no hands' riding?
38) Determine the maximum range of audibility of a conversation in open air.
83) Estimate the time for a pond to freeze.
89) Estimate the thermodynamic efficiency of the firing of cannon and hand-guns.
127) Describe the electrical effects caused by the earth's magnetic field when water having an electrical conductivity flows in rivers.
200) Two satellites have a head-on collision. Describe the subsequent events.
202) Estimate the height to which a person can pole vault. Determine the cross-section of the pole.
210) A student comes late into a lecture-room. She is wearing a strong perfume, Estimate the time that elapses before the lecturer can detect the scent of the perfume.
One could argue that almost all of our highest voted questions are of this same spirit. (e.g. , , , , ) In fact, the recent mass of a coin question serves as a perfect example of a Kapitza style question, the question itself being rather short and ambiguous. The answers really embraced the nature of a kapitza question, and building off one another ended up with a fairly compelling analysis. But, the question itself was poorly received (initially), as discussed in this recent meta post. There, the community points out that the question is not a good question, as we currently define a good question. Other seemingly valid 'kapitza' type questions pop up and are closed as too broad or off topic or engineering fairly regularly, and probably rightfully so, given our current definition of a good question. This search returns an approximate list of such candidates, some examples that stood out to me include: , , , , 
This brings me to the actual proposal. The proposal would be to dedicate a new tag, either 'kapitza' or 'challenge' to mark these kinds of questions. Simple, ambiguous, open ended questions without prior research, but in an attempt to try to foster this kind of collaborative, quantitative modeling challenge. True 'kaptiza' questions would be evaluated on the criteria Kapitza himself outlines in the introductory pages of his problems. So, much like our current homework tag, the rules for 'kapitza' questions would be a little tweaked, the questions would be evaluated by a different metric than standard questions. The goal of the question being not to find the one authoritative answer as the questions are designed not necessarily to have one, but to really challenge our contributors to try to collectively create interesting content introducing a wide array of physics topics. Think of it as a sort of codegolf.SE section for the physics.SE, but much more pedagogical.
In practice, this would require little else than a dedicated tag and explanatory wiki page (and corresponding evolution in community behavior associated with the tag). For the tag name, I would volunteer 'kapitza', a unique tag, to mark the questions as similarly unique, but with the disadvantage that it doesn't mean anything unless you know what they are, or take the time to read the tag wiki.
I believe this new tag would be a way to codify some of the observable trends we have already, namely the kinds of questions and answers that people like to upvote, making it easier to discover and contribute to other, similar questions. I think this would help also to alleviate some of the tension we have between some questions getting closed early as too broad, while some similar questions becoming our most highly voted. A dedicated tag would also enable those uninterested to block the tag from their view.