If the matrix is a reflection matrix, what happens when it transforms vectors in the reflection plane?
If you know 2 vectors in the reflection plane, can you find the plane?
If you transformed v into v', is v + v' in the reflection plane?
2 non-parallel vectors is enough to determine the reflection plane (since the plane needs to contain the origin (0,0,0)).
The plane consists of all points that are the linear combinations of the 2 vectors.
Pick two different vectors v, w. Transform them into v', w'. If v + v' and w + w' are two non-parallel vectors in the plane, then you have your plane. If they happen to be parallel, you can also use this to determine the plane in another way (draw a diagram).
I believe you could also find an eigenvector associated with the eigenvalue -1. This would give you a normal to the plane, because if you reflect a vector that's normal to the plane about that same plane, the resulting vector will be -1 times the original vector. That's how I've been taught to solve this, but I think snowtea's way might be easier.