Start by getting the equations of motion using Lagranges equation. Here is a reference that saves me the trouble of doing it: (Example 3.2)
Schaum's Outline of Theory and ... - Google Book Search
Or you could attack the thing head-on.
Newton's Law in the two coordinate directions says
$\displaystyle m \frac{d^2x}{dt^2} = -\frac{x}{\sqrt{x^2 + a^2}}$
$\displaystyle m \frac{d^2y}{dt^2} = -mg - \frac{a}{\sqrt{x^2 + a^2}}$
That system looks horrible to solve, but the thing is that I can't come up with a good generalized coordinate for this problem. The best I can offer is to put an angular variable at the focus of the parabola.
However, things aren't all bleak. All we need is the coefficients of the expansion:
$\displaystyle \frac{d^2x}{dt^2} = A + Bx + Cx^2 + ~...$
Presumably if h is small then A = 0, the coefficients of the powers of x higher than 1 are small, and B is negative. In that case
$\displaystyle \frac{d^2x}{dt^2} \approx Bx$
which is the harmonic oscillator equation. I leave it to you to check that these conditions come out. (I haven't checked.)
-Dan
Edit: After seeing the original posting of this problem it is clear that the problem is requiring an exact, not approximate solution. The only way I can think of to get this is to use the Lagrangian technique. But again, I can't think of a good generalized coordinate here.