Thinking outside the box in a way that actually improves upon things.
Or it could be an entirely new path, in which case the genius of it all is apparent.
Nurture such thought processes... Well.
I think that real geniuses - not savants - are hugely creative individuals.
Some of the most intelligent people I know of are incredibly well-spoken.
I'd say that a lot of geniality can back-traced to how strong the language centers of your brain are (Broca's area and Wernicke's area).
If you have the correct grasp of words so that you can express yourself and your thoughts in a fashion that does not restrict you...
Well, then you've come far in the thought process already.
Let's take the example of Helen Keller. Until she was in her early teens she did not know a word.
Before language, her only memories are of primal urges and base feelings like hunger, anger, satisfaction.
After language was taught to her by Anne Sullivan and she could start teaching herself things, it was a whole new world.
Thus, I reason that higher proficiency in one main language and many auxiliary languages gives you better outside the box thinking,
because different languages contain different nuances and meanings in words, which incorporates entire ways of thinking.
When you see a larger picture than you can using only one language, thought comes easier.
Of course, a person needs a good sense of logic and many other things as well; like this little thing called creativity.
Those two seem hard to mix, though. In all honesty, I think it's a matter of human evolution and or possibly nutrition/environment/upbringing/education.
Some have it easier from the start genetically, and they are few and far between. Even fewer get the right conditions for advancing.
But, all in all, i'd say the geniuses that I have met are linguistically gifted no matter what field they are in.
Not saying I am a genius myself. That'd be boastful, since I haven't accomplished anything worth noting.
I agree with Disregard that the opinions of other people are indeed important, as well.