What is the basis of your use of the word "mind"?
The definition provided for "unconscious" was the complement of a set, but it sounds like the universe under which the complement is being taken needs to be refined. "Unconscious mind" is a common term, so it's offered to provide that fence. That's the
basis for the use of the word mind.
But the
use of it in a particular way is simply something I'm leaving open (primarily because, as I understand, there's no one conventional use of the word mind; equally important, because I'm not trying to advance a particular use). I'm not sure I care, so long as the word "mind" seems to be appropriate to apply to whatever category you carve out. The appropriateness test does have requirements though, so I'll just outline a sketch of what I think should apply:
- It should deal with categories of things that we are conscious of. Generally, if we can name it, it's a reasonable category.
- It should in some fashion work the same way with these categories when we're not aware as it does when we are (however we connect the dots, it should use at least some similar dot-connecting capability).
- The category-mapped workings of the unconscious part should somehow be related to the categories we formed consciously; ideally, it should be causally related (such as the concepts we can access consciously being available to it for use; or its potential to alert us consciously with the same concepts, as the case may be for this particular thread).
Given these constraints, there are a number of processes you might want to include or exclude--knock yourselves out! Just tell me what the final sentence is and what you mean by it, and I'll start from there. I reserve the right to apply a "utility" requirement, which is roughly that the concept fencing itself plays an important role in the description; or that it contributes to a general theory about the entire fencing; or that the fencing as a whole incorporates an object that has a lot of interesting properties to the exclusion of what it doesn't incorporate.
Although consciousness is a totally subjective experience defying the objectivity needed for scientific analysis, circumstantial evidence (to borrow a legal phrase) seems to indicate that the highly symbolic nature of human language is the basis of human memory, consciousness and thought. See Peter Carruthers, Michael Dummett, Benjamin Lee Whorf, et al.
I'm skeptical; I've no doubt that human language, human memory, human consciousness, and human thought are
correlated, but I think it's possible here that there might be a confounding factor behind these correlates.

I'm not alone here either (see Churchland and Churchland, Pinker, Minsky, et al).
To highlight a few reasons to be suspicious:
There are animals who are quite adept at problem solving, for example, and for some categories of problems they humiliate us with their abilities (just try beating a chimp at short term memory sorting of randomly scattered numbers). There's evidence that babies start thinking before they acquire language.
There's the fact that I'm spending so much time simply trying to explain these objections that I'm conscious about in a clear language. That suggests to me that perhaps language is generated from thoughts, not the other way around.
Or consider specific thoughts in the face of ambiguous language constructs; in other words, given the observation that a particular phrase can have multiple meanings, there must be a many-to-one mapping between a language construct and a thought. That's not quite a desirable property for an argument advancing that language possibly is a key behind thought; it seems to suggest the relation works in precisely the opposite direction.
Feel free! However, if you do have anything other than anecdotal claims, I would be interested in learning something new. In any case, you should ask those people here who have made claims about "unconscious thinking or problem solving," since I remain sceptical of such claims.
And here's where I jump back onto your side for a while. With the facts in, the subjective reports of experiences with "my unconscious mind solving the problem for me" seem to involve subjects reporting on the subjective mechanisms of a mental state that by definition they are not privy to.
There is the: (a) I had a problem and was stuck, the (b) I went and did something else, the (c) suddenly I had a Eureka moment, and the (d) the solution presented itself to me. I can buy that (d) was the solution to the problem, and that by (a) the problem may have been difficult. I can accept that (b) the problem was shelved, and (c) there was at some point a Eureka moment. But from the report, all I can conclude is that a surprise occurred, a conscious review of the problem due to the surprise was afoot, and that the result was the solved problem.
The explanation offered is certainly based on current lore, but since the workings of the unconscious are something we're not privy to, it seems a bit premature to conclude that there was an actual processing of ideas that led to observations (a) through (d). Other explanations of these points are possible, and should be ruled out; for example, the conscious review of the solution at point (d) may actually be where the problem solving occurs, and the Eureka moment at (c) may simply be a successful recall of a relevant piece of information. Though it's possible that there was an unconscious processing of information analogous to our conscious processing and that this solved the entire problem, it would seem that it is a thing that should have to be established.