What are some good ways to get children excited about math?
Solution 1:
Graph theory! It's essentially connecting the dots, but with theorems working wonders behind the scenes for when they're old enough. Simple exercises like asking how many colors you need to color the faces or vertices of a graph are often fun (so I hear). (Also, most people won't believe the 4-color theorem.)
Solution 2:
I recently taught a once-a-week geometry class to grades 7-10 where we did some graph theory, some surfaces, some spherical geometry, a lot of complex numbers, and basic ideas of homeomorphism and homotopy. Most couldn't work with complex numbers at the beginning: showing their use in proofs of analytic geometry results had the benefit of not boring the kids who had already seen and worked with complex numbers, and giving the kids who hadn't a reason to learn the technicalities (getting under the hood of these pretty pictures).
Euler's formula and the relation of planar graphs to polyhedra showed them the basics of a connection between geometric and algebraic intuition, outside the coordinate geometry many were used to, and got them thinking about ways to define familiar things mathematically (space), and what we can learn about familiar things through mathematics (surprising things like orientability through the Möbius strip, or sphere eversion).
In general I've found geometry to be a very good place to start with people with a professed fear or disinterest in mathematics. For me, it's the quickest way to show the distance between what mathematicians play with when they're doing math and what was taught in high school, with the two column proof nonsense and the treatment of math as a branch of formal logic.
Of course, I suspect that I find this path the easiest because it is one of the ones I am most excited about in mathematics, and I further imagine that in many cases, the best thing to use to get people excited about math is something you yourself are really excited about, as long as you can think to translate it well.