Just an opinion: ideas do not come from nothing, so the larger the data pool (memories, experiences, interests) the more ideas are likely to be generated.
It very much seems like we live in an age of hyperspecialization; people know very much about relatively few things. Generally, these areas of knowledge are complimentary or related. Sometimes they overlap outright. Life is barely long enough to get good at one thing, so people often choose to specialize early and stay on one very fixed path.
From the outside looking in these collectives of experience are very tribal. They develop their own languages and symbols. They become these closed systems where ideas aren't created as much as they are simply refined; bounced back and forth among tribal members. But this does not seem to be a very good pattern for long term growth or sustainability. Homogenization leads to extinction.
What I mean by that can be understood by looking at the evolution of life on Earth as an example of the obverse. Evolution tends towards diversity. Diversity gives life its best possible chance of success. That way, when an asteroid slams into the planet, not everything dies. If evolution had tended towards homogenization (only making the best dinosaurs possible) instead of diversity, the K-T Event might have turned this planet into a floating rock.
It may be a bit of a weak analogy, but I feel like the same principles might apply fairly well to specific areas of knowledge. Ideas are the mutations that allow knowledge to change and evolve into something new. Exclusivity and specialization are a sort of homogenization that leads to stagnation and fewer truly new & good ideas. Not that ideas don't happen at all, just that maybe they happen less often than they should... or could. I don't know, really. This is mostly just speculation based on personal observation and opinion.
Colloquially, I can say that the people I have known in my life who seem to have the most ideas are the ones whose interests are all over the map, so to speak. They tend to be older, with a deeper well of experience to draw from. Their knowledge pools, being varied as opposed to complimentary, allow them to look outside these otherwise closed systems and make inferences, or to see patterns that people too mired within the subject matter might easily miss.
They may not always be good ideas, but they are often striking in their seeming originality and unexpectedness.
An example that comes to mind is of a family friend who worked for years in automotive manufacturing before going back to school to get his certification as a laboratory technician. He got a job as a lab assistant at a University research hospital. He would overhear the researchers in the break room talking about their current projects, and one of them that really grabbed his interest was the problem of infectious disease control measures, specifically, getting healthcare professionals to wash their hands between patient interactions. He had the idea, based on his experience in manufacturing, to apply Poka Yoke (a Japanese manufacturing term that roughly means error-proofing) to the problem of getting nurses and doctors to wash their hands between patient encounters. His idea was to install sink-locks at all the doors to patient rooms. These doors would only open from the outside if the sink was used for at least 20 seconds immediately prior to opening them, or if an emergency button was pushed. From the inside they open at will. He mentioned the idea in casual conversation with one of the senior researchers who was so excited by it that he wanted to design a study around the concept.
I feel like there is a potential benefit to be had by looking outside as opposed to focusing too intently within. Maybe spending some percentage of time learning about completely new things as opposed to only endeavoring to learn new details about things we already know might yield an increase in new ideas. There's nothing wrong with getting out of our comfort zone and challenging our perspectives.
Starting from the assumption that this is learned("Nurture"):
Generally, through the ambient feeling of your experiences, learning to see yourself as someone who can find solutions vs. learning that other people have the answers. This would be a very tricky edge to ride, since often other people DO have the answer (experience, expertise, whatnot). More on that below, first just a few examples:
Imitation approach: I suspect that being exposed to role models who invent and think will help anyone. This might even offset negative reinforcement directed at oneself to some degree (In stories, the hero(scientist etc) can also meet adversity.)
Reinforcement-explanation approach: Experience of having your ideas shut down, ignored, etc. Being punished for generating ideas would leave two options: Stop or at least shut up about it. Alternatively the "support" should obviously not look like endless praise, but like interest and questions/suggestions where more information might be found - unless of course something is clearly corect/incorrect, in which case that should be said. Positive reinforcement here would have to aim at the process of investigation - and getting closer. No meaningless fetishizing of the search.
Never getting anywhere/learned helplessness. Thinking is hard, aandhaving that pay off too rarely (that may differ from person to person) will teach us to avoid the trouble in the future.
Picking up from above: I find that learning something is often more or less tricky, depending on how easy it is to formulate an ideal that can be "blindly followed". When someone says "just train/read/empathize etc MORE" - implementation may still be difficult, but the mindset is simple. You only need to be a maximizer. When you start adding "don't be too hard on yourself/don't overdo it..." - the problem is hidden in the implied but unclear "too much".
With many activities the solution may be in understanding the goal and the process properly. Sticking with the examples so far:
Training requires rest, nutrition etc to be effective. Treat adhering to the dosage that's right in your case (you can consult professionals) as part of the discipline, not as breaking the discipline. If you injure yourself, that's inefficient training. Reading til you damage your eyes, compromise your sleep etc. is not efficient if you actually want to learn. Don't empathise to the point where you can't distinguish between other and self. That's not empathy, that's confusion.
In such cases it's "don't fetishize the process, but remember what you're trying to accomplish, and redefine your approach such that it implies everything you have to take into consideration".
This is often tricky, but once done, we can, in principle, max out again, which is cognitively easier.
With regards to problem solving/actually thinking own thoughts I think it's difficult, because firstly you have to be able to:
It's the same problems, only as other skills/habits, only seeing the problems is more tricky, and there are no coaches you can rely on to tell you exactly how much of something is too much, and how to redefine your approach. The best I have heard is what I call Eliezers "winning" approach. Ask in every instance how it may be better done, and what is required to MAKE IT HAPPEN vs. going through all the correct motions/procedures.
Asked in this fashion the question of why this skill is rare seems to answer itself. It's very hard to learn - for more reasons than I named now, of course.
Starting from the assumption that this is an innate ability thing ("Nature") I can't see anything beyond "Well, it's a raw brainpower thing". This seems anecdotally refusable, so I'm not in this camp. Nonetheless more smarts helps, of course - not least in developing a positive attitude towards problem solvin, but obviously just in terms of "resources" - and maybe if we enhance people's intelligence we'll see more problem solvers? I don't know. I find it hard to see why people wouldn't just use their greater intelligence to prove they don't need to change anything, if no "nurture" elements come in and help.