Can one incubate on a specific idea in one's sleep? In this question it is described that intentionally not thinking about a problem can allow the subsonscious to find an answer to the problem; this is known as incubation. I have heard and read it a few times that if you want to solve a tough problem you can read/think about it just before going to bed and let the brain come up with a solution while you sleep.  (I can find no references for this at the moment, however. Sorry)

Also, there have been few cases that provided a solution during sleep. (Benzene structure is one which I remember) Is it mere chance or false advertising, or is the brain really capable of incubating on an idea during sleep?

A robotics guy I saw on TED claimed that he used to keep a notebook by the bed, so that he could jot down the ideas if any came to him during sleep.

Can one's brain incubate on an idea while sleeping? Can one train one's brain to incubate on a specific idea during sleep?
This study found that REM sleep "enhances the integration of unassociated information for creative problem solving", and to an extent that surpasses the benefits of "quiet rest and non-REM sleep". try a google scholar search for 'psychology sleep  incubation '. 

Interestingly, the experiment that yielded the first discovery of a neurotransmitter was famously claimed to have been conceived of during a dream and was conducted during the middle of that same night. The scientist, Otto Lowel, had failed to successfully document the same dream nights prior.