I believe this is apocryphal. Watson likely said this because he stole Rosalind Franklin's research.
This is probably something I need to research more, because the questions scientists are dealing with are too deep to ignore the one big question, and I wonder who struggled with it.
Mind you he was helped by a nice coincidence. Franklin knew all 230 space symmetry groups, and so had to sort through them. Watson only really knew one - his PhD thesis was on a protein with the same group as DNA.
It's happened several times that I struggled with a bug for hours, then suddenly came up with a key insight during the commute home (or while taking a shower or whatnot), while not actively thinking about the problem. I can't explain this any other way than some kind of subconscious "brainstorming" taking place.
As a side note, this doesn't mean the hours of conciously struggling with the problem were a waste. I bet that this period of focus on the problem is what allows for the later insights to happen. Whether it's data gathering that alows the insights to happen, or even just giving importance to the problem by focusing on it. Most likely it's both.
(n.b. historically speaking in the literature, "subconscious" was the word used to describe Jung's "woo-woo" ideas about a collective subconscious shared across populations, and unconscious was the word used for ideas about a single brain a la Freud which is what we are talking about here)
For me, the subconscious is the wellspring of sudden insights.