Did Euler have a trick? Did Euler have a trick for discovering things? Some sort of general method he could apply to mathematical objects he came across to see if they yielded any new truths? Did he just ask the right questions at the right time? Was he lucky? Did he have a very advanced and well-developed intuition?
How would you train a kid to become the next Euler. Just imagine you were given a kid who was really good at math, and it was down to you to give him all the training needed so he could embark on a career like Euler's (and you are guaranteed he would work as hard as Euler). For example, if I worked all day exploring maths I probably wouldn't discover a quarter of what Euler discovered, and even then I'm being HIGHLY generous to myself.
How did he do it?
 A: I think (though I am not completely sure, nor could I be) that this question is predicated on a basic fallacy: that a good way to become a genius in field $X$ is to find person $Y$ who is/was a genius in field $X$ and try to emulate them as a person (or even as a practitioner of $X$) closely enough.  
There is a certain ineffability to genius.  It is interesting to try to "understand" it, but difficult to do so; but moreover it is not clear that such understanding is the key to replicating it.  Pete Sampras was one of the genius tennis players of the modern era.  I think most serious tennis fans could say a lot about what made him so great.  But although most players could probably find something in Sampras's game that would improve their own, the young tennis players who are the best among their cohort did not get that way because of the fidelity of their emulation of Sampras.  Excellent young tennis players have certain qualities that Sampras had (and probably certain qualities that Sampras didn't have), but they get those qualities through a combination of inherent gifts, lots of practice and conditioning and participation in the game: i.e., a big part of being a great tennis player is competing against other tennis players and learning/responding to them.
I think it is similar in mathematics.  As a mathematician Euler was admirable in almost every way I can think of.  One can learn a lot of mathematics by reading his work.  However nowadays one could probably learn mathematics more efficiently -- and certainly, learn more mathematics of contemporary interest and importance -- by following more recent and then contemporary mathematics.  In order to be a great mathematician you have to be very bright mathematically (but I fear this may be a tautology) and very hard-working, but you also have to immerse yourself in the mathematical community (at least at the level of reading papers; I don't necessarily mean social interaction, although for most of us that helps a lot too).  
I honestly think that too close an emulation of Euler is a waste of time for an ambitious young mathematician.  I am confident that Euler had his tricks (we all do...) and that by careful reading of his work one could learn those tricks very well, probably better than most contemporary mathematicians.  But that's not the route to being a genius mathematician: you need to find your own tricks.  Yes, you will find inspiration in other people's work, but not just one person, and the hard part of it is the synthesis of past work with your own ideas.  
Further, I think that if you studied Euler carefully enough you would probably find that he is "smarter" (quicker, more immediately insightful, better intuition and more accurate calculation...) than you and also "harder working" (more hours of work every day; a willingness to bang your head against the hardest problems rather than being content with doing something nontrivial; more institutional support for a very mathematics-heavy life...) than you are, and this will be true almost no matter who you are.  So close emulation of Euler might actually be discouraging.  
A: Robin Wilson wrote about Euler (see comment):
Leonhard Euler was the most prolific mathematician of all time. He wrote more than 500 books and papers during his lifetime — about 800 pages per year — with an incredible 400 further publications appearing posthumously. His collected works and correspondence are still not completely published: they already fill over seventy large volumes, comprising tens of thousands of pages.
Euler worked in an astonishing variety of areas, ranging from the very pure — the theory of numbers, the geometry of a circle and musical harmony — via such areas as infinite series, logarithms, the calculus and mechanics, to the practical — optics, astronomy, the motion of the Moon, the sailing of ships, and much else besides. Indeed, Euler originated so many ideas that his successors have been kept busy trying to follow them up ever since; indeed, to Pierre-Simon Laplace, the French applied mathematician, is attributed the exhortation in the title of this article.
