I suspect there is a lot of truth in that phrase. And that in many cases problems sit at the back of inventive minds, waiting for eureka moments.
It is important for inventive minds (which I believe all minds have the capability to be) to be aware of the worlds problems. Of which there are many.
How can we ensure the awareness of those problems? The biggest challenges are often discussed in the media, for example reliance on fossil fuels and anthropogenic climate change. But not all problems are promulgated in this fashion.
I propose an online project for listing, describing, categorising (hierarchical tagging), prioritising and structuring the world's problems. If such a project could harness the community spirit of wikipedia and crowdsourcing techniques, and I believe it would be a valuable resource for inventive minds.
I believe that one of the greatest challenges of the current era is in maximising the capability of the human mind. Through education, through tools that compensate for the mind's deficiencies, and eventually through medical and biotechnological techniques.
An afterthought: if invention is driven by necessity, perhaps art can be described as inventiveness that is not driven by necessity.