Here's a method which will give you the nearest vertex of the gameObject's mesh. I've written this code to be suitable for putting into a MonoBehaviour script, because it uses references to the gameObject's "transform" and "MeshFilter" components.

An alternative would be to make a static "helper tool" version of this function which could accept both a point and a GameObject parameter.

public Vector3 NearestVertexTo(Vector3 point)
{
    // convert point to local space
    point = transform.InverseTransformPoint(point);

    Mesh mesh = GetComponent<MeshFilter>().mesh;
    float minDistanceSqr = Mathf.Infinity;
    Vector3 nearestVertex = Vector3.zero;

    // scan all vertices to find nearest
    foreach (Vector3 vertex in mesh.vertices)
    {
        Vector3 diff = point-vertex;
        float distSqr = diff.sqrMagnitude;

        if (distSqr < minDistanceSqr)
        {
            minDistanceSqr = distSqr;
            nearestVertex = vertex;
        }
    }

    // convert nearest vertex back to world space
    return transform.TransformPoint(nearestVertex);

}

If you really need to find the nearest point within the surface of the nearest triangle of your mesh, this code could serve as a good starting point.

Don't know if you're still interested in this, but I've posted some code for this on the forum:-

http://forum.unity3d.com/viewtopic.php?t=36772

It's a bit like the code Duck suggested, but it also implements the nearest point on triangle part and uses a spatial data structure to store the vertices so as to reduce the overhead of the search.

As a quick workaround, you could try the following in whatever script you're calculating the point from: -Un-rotate the object with the collider so it's rotation = Quaternion.identity. Keep the previous rotation in a temp. variable. -Use the Collider.ClosestPointOnBounds method to calculate the Point -Rotate the object back to it's original rotation. -Rotate the point along the object's position with the same rotation you applied to the object.

This should, in theory, result in a point that's on the object/collider's surface even when it's rotated.