As far as I know "gather way" means "to attain headway" in navigation. So I believe this is a kind of idiom or something like that. I just found this sentence with no context, so I don't know if you could help me with the meaning of it.
As you already know, "to gather way" is a nautical term that means to begin to move or to move with increasing speed. (To attain headway.)
Without context (you said their wasn't any), I have to imagine that it is perfectly natural for someone in my own boat club to use the phrase "the idea gathered way" to mean that the idea picked up momentum.
The notion that ideas have momentum is not unheard of, especially among people who pay a lot of respect to the process of ideation. Here is one example, in an article titled "The Momentum of Ideas", taken from the New York Times quoting Newt Gingrich:
I’m a very idea-oriented candidate. And I think the Internet makes it possible to create a momentum of ideas that’s very, very exciting.
Another metaphor similar (but not identical) to "momentum of ideas" might be "avalanche of ideas". These bring to mind that the ideas, once they have been conceived, have an unstoppable power, being able to move of their own accord.
The phrase gather way is normally used with ships as suggested by OED:
gather way (of a ship) begin to move.
I haven't come across the use of the expression "idea gathered way" anywhere before. A search in Coca and Google Books would suggest that nobody has use this expression in any book or magazine. Therefore I don't think gathered way should be treated as a phrase that functions as a single unit in the expression "The idea gathered way". It is most likely that there are some words missing after way, which would mean it is merely an extract from sentences like "The ideas gathered way too fast...".