Not exactly an answer to your question, but I think relevant to the discussion. Also, I make a disclaimer now, much of what follows is my opinion based on what I've seen. If there are mathematicians or anecdotes that you think would give a better perspective on the situation, I'd love to hear about them.
It is worth noting that Euler's initial proof of the fact that $\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^2} = \frac{\pi^2}{6}$, whilst elegant, isn't rigourous by today's standards. It showed great intuition though, and that's how the sort of discoveries you are talking about are often made. A relative newcomer to mathematics has an insight which leads them to make some progress on a problem. However, their intuition often leads to non-rigorous arguments. With some luck, the argument can be made rigorous by those with a larger mathematical toolbox.
Another example of such a mathematician is Srinivasa Ramanujan, an Indian mathematician who made many independent discoveries with almost no tertiary level mathematics education. From the Wikipedia article:
"The limitations of his knowledge were as startling as its profundity. Here was a man who could work out modular equations and theorems... to orders unheard of, whose mastery of continued fractions was... beyond that of any mathematician in the world, who had found for himself the functional equation of the zeta function and the dominant terms of many of the most famous problems in the analytic theory of numbers; and yet he had never heard of a doubly periodic function or of Cauchy's theorem, and had indeed but the vaguest idea of what a function of a complex variable was...". When asked about the methods employed by Ramanujan to arrive at his solutions, Hardy said that they were "arrived at by a process of mingled argument, intuition, and induction, of which he was entirely unable to give any coherent account." He also stated that he had "never met his equal, and can compare him only with Euler or Jacobi."
I would imagine that someone looking to make such discoveries today would start in the areas that Ramanujan worked in; whilst they are incredibly deep, they contain many results which are easy to understand. For example, an infinite sum for $\dfrac{1}{\pi}$ is easy to understand, even though it may be incredibly hard to find.
The last point I would like to make is that while Euler's discovery was actually the solution to the well-known Basel problem, Ramanujan's discoveries weren't in the form of solving open problems, at least, not any that he was aware of. Ramanujan worked in isolation and made amazing discoveries by exploring mathematics, letting his curiousity and intuition lead him. I would encourage a modern day Euler or Ramanujan to do the same; the brilliance such a person would possess may allow them to answer questions we have never even thought to ask.
