Solution:

By the product-to-sum identity, $2\cos(k^2 a)\sin(ka) = \sin(k^2 a + ka) - \sin(k^2 a - ka) = \sin(k(k+1)a) - \sin((k-1)k a).$ Summing over $k = 1$ to $n,$ the terms telescope, leaving $$\sin(n(n+1)a) = \sin\frac{n(n+1)\pi}{2008}.$$

A sine is an integer only when it is $-1,$ $0,$ or $1,$ that is, when its argument is a multiple of $\frac{\pi}{2}.$ So we need $\frac{n(n+1)}{2008}$ to be a multiple of $\frac{1}{2},$ i.e. $1004 \mid n(n+1),$ where $1004 = 4 \cdot 251$ and $251$ is prime.

Since $n$ and $n + 1$ are coprime, $251$ must divide one of them, so $n \ge 250.$ For $n = 250$ the product $250 \cdot 251$ is not divisible by $4.$ For $n = 251$ the product $251 \cdot 252$ is divisible by $4 \cdot 251 = 1004.$ The smallest such $n$ is $251.$