Often we are asked, "What's your opinion about this?"
Hopefully civilization will advance to the point where this question is regarded as silly and uncouth and on a par with, "Hey, who just farted?"
I propose a much better question to be used in its place:
"How will you work out what your opinion is?"
Yes, indeed. But until this becomes compulsory, I think you will just get blank stares. When one is asked for their opinion, usually we are at the point with something where some thinking is necessary, and for that, most people expect an opinion (which may or may not correspond to some thinking done - usually not done). Taken at face value, your question asks that someone spell out the reasoning behind the opinion; this is OK, given enough time (and interest) exists in the circumstances. You are asking for a holdup - whereas others may just want to get something dealt with.
I understand only too well why your question is superior, therefore, try it out and see how you go.
A case in point is the recent announcement by CERN of the "discovery" that certain neutrinos go faster than light. Absolutely everyone in the science world has an opinion about this and by now everyone has voiced his/her opinion about it. I have never, in all my born days, seen scientists rage at one another in the emotion-ridden manner they adopt when addressing this issue with each other. It seems that the way this particular game is played is to denounce the thinking of anyone whose beliefs don't accord with yours as mere opinion, unsupported by facts. The very phrase "mere opinion" says it all; even though this is all anyone has to offer at this point in time, nobody will personally admit to their deep emotional need for Albert Einstein to be right about everything, nor will anyone equally confess to really wanting the CERN result to be true and all the text-book ripping-up that will be the consequence of that.
Perhaps the problem is with the term "opinion". On one hand we have "mere opinion" which seems to belittle the whole notion of what having an opinion is. On the other hand, there are those who claim to have "an informed opinion" which somehow seems to rehabilitate mere opinion to the point where we shouldn't just put up with it, we should go with it, since it probably emanates from an expert.
Opinions have to be formed somehow. Unfortunately, an "opinion" is pretty much a judgement or an assessment of something so tends to be the final word either from that party, or in the whole case, where the opinion was the thing sought. My problem with "informed opinions" is that I instantly want to know what is informing the opinion. If at all possible to ascertain, it usually turns out to be prejudice, or, if that seems too heavy a term "bias" or "preference" might do; the important point to realise is that all opinion is based on a certain perspective which is why all opinions - informed or otherwise - are highly fallible things.
"What is your perspective on this?"
"What is your impression of this?"
"What do you take from this?"
"From where you are standing, what do you make of this?"
"Can you trial a summary judgement of this for me, if you think there is enough to go on?"
"I am collecting opinions about this. What is yours?"
In each of these, the questioner is delimiting the value of the exercise as one would any work in progress. At no point should we ever get the impression that the thinking tap has been turned off.
Except for "trial summary", all the questions imply that the respondent can and should have a pre-formulated opinion which doesn't need to be developed further.
Perhaps these two are an improvement on "What is your opinion?":
"What do you think of this?"
"What do you make of this?"
... which both imply movement, though neither really pushes the respondent into seeing that it's their responsibility to put themselves into a position where they can see the issue clearly.
The question "What is your perspective on this" is not a request for a pre-formulated opinion (although, with most people that's probably what you will get back). It's a request for someone to lend you their perception for a moment. Your perspective on something is really your "viewing-platform" - an honest response to such a question would not be so much an opinion as a report on the content of perception vis a vis such-and-such. This is a little like putting on your hW and the kind of thinking that doing that entails. The same goes for the question "What is your impression?" An impression is literally that: the imprint that something leaves on something else; in this case on your Special Memory Surface. This is all first-stage perceptual thinking: data-gathering if you will. Opinions are much more than that - as you are clearly suggesting.
There is also the possibility that the person asking someone else to adopt responsibility for the formulation of their opinion is themselves behind the eight-ball when it comes to understanding the issue clearly. This may be why they asked for someone else's opinion or perspective or whatever, in the first place: a genuine need for assistance. Under those circumstances it mightn't really serve your purpose to doubt the credentials of their opinion! The context, as always, is critical.
You make some very good points.
I agree that peoples' perspectives are part of the landscape and that it is OK to ask for them as such.
Nevertheless I believe everyone should understand that they can think of better opinions without getting more facts.
Sure. But isn't the big hurdle here the fact that people are rarely trained as thinkers, yet everyone feels entitled to an opinion? I'd go so far as to say that the voicing of opinions and the trained-up nature of the thinking behind them probably stands in inverse-proportion. The better the thinker, the less likely they are to opine - unless they announce they are wearing their hR.
Your asking them to show the research or the thinking they will do before they arrive at their opinion may well kill off their willingness to offer it, particularly if they know you are a trained thinker and they aren't.
Nevertheless - your core concept here is the need for more hH (that's Blue Hat) meta-cognition, or, thinking about thinking and we certainly do need to encourage that.
Perhaps you might design a hypothetical exchange between two people where someone responds adequately to your request for some hH on how they might arrive at an opinion over something.
"I am collecting opinions about this. What is yours?"
I think, Kim, that is the question you are trying to avoid in favour of something less unjustifiable as you have described 'opinion' - At least to my understanding
I thought you might not entirely approve.
But that's not the only reason I wrote it.
Luis: Question Re-phrasing: How will you work out what your opinion Will Be?
That seems worth it. It suggests that the person asking this question is not actually ready to receive an opinion at this time, because people generally, are all too ready to offer opinions. Opinions are rather cheaply made, unfortunately. Any opinion you want to give me on the spur of the moment I will accept only as an "impression" or perceptual report, as I like to put it. I will put that on the table and look at it but I am not going to use it as a judgement. You are invited to process your perceptions into a fully-blown opinion if you can, but I will accept it only on the basis that you can tell me how you arrived at it. This may well take time. That seems to be the pact Phil is setting up. Good wine takes time to make. Changing from the "is" to the "what can be" is always the sign of real thinking being done. Excellent, Luis.
I've thought about this question quite a bit, because the way someone thinks has to do with how they respond to move. I teach people to move easier, so I want to know their thinking (and "self-talk.) I also want them to be able to witness it as it's happening, without passing judgment on themselves as they are doing the thinking. Some of what I do when I ask people to show me how they're working out their thinking - I explain it's like the teacher asking the student to "show their work."
In an attempt to both be personal and to ask the question without being confrontational, I say such things as...
Please talk about what you're up to at this moment...before you go into action. how are you approaching it? What are you saying to yourself in your thoughts as you go into action?
I'm curious how people work out what you're intending to do, ahead of doing it - how is that happening with you now?
The other point is that if the questioner "goes first," not only does the other person have a better idea of what is being asked...but they are less threatened that if they reveal their thinking. There's a fear that one's thinking will be pounced on as being wrong in some way.
There are really interesting answers to these sorts of questions. Evidently, finding out what their music students were thinking about revealed some misconceptions people have about the way they are structured as humans. These misconceptions caused their students to move in funny, stiff ways. Getting a little education about structural anatomy really helped to eliminate some of the awkwardness in what was demanding excellence and continuing stamina - playing music professionally. Here's a link to a preview of this, illustrated in a DVD for sale made by Barbara Conable at http://movewellavoidinjury.com
How will I work out what my opinion is?
I don't know that I will
My opinion is fluid ... it changes with time and experience
Cities and opinions are analogous to living things in many ways and it's a bit like asking when New York will be finished - If it ever achieves zero growth, declines and disappear forever then it will be finished ... dead ... and not before
In some ways I feel this question to be a bit like the stock interview question "What's your greatest weakness?"
It's a non-question that cannot be answered
What my greatest weakness is depends upon whom you ask ... and I am annoying to as many different people for as many different reasons as there people to whom I am a disappointment in that specific regard
Even if one individual could specify my greatest weakness as being of more significance than any other the last person in the World to identify it would be me
So, in much the same way, I am uncertain how I will form my opinion
Do I even have free will? 
I believe I will form my opinion based upon a combination of reference to personal experience, the learned wisdom of others and my own moral values - I may however be a hypocrite and do no such thing
But, I like to think of myself as being as nearly impartially Soloman-like as it is possible to get without being a transubstantiated parable ... and form it a I have suggested
But as another more fruitful approach than mere enquiry ... which, as you have suggested is sui generis limited in scope ... I would suggest a three-tined  approach:
1) What is your opinion on this?
2) How did you reach it?
3) What would make you change it?
 I do, as it happens, have an opinion on that matter ... but I'm not sure it's mine ; )
 You might not get another chance to land the blow, so ... when you do ... you should take advantage of the opportunity to stab them in the eye with a fork - It'll keep them off balance and their thinking more creative as a result ;)
Another useful opinion question might be "And what will it be in five/ten years' time - And why?"