Consider a box centered at some point in space. Imagine a line from the center to the top-right corner. This length is effectively the hypotenuse of a triangle formed from taking the center of the box and drawing a line right and then up. These sides of the triangle are half-extents. In the picture, the red lines are the half extents and the blue line is the hypotenuse -- the distance from the center of the box to the corner. Call this blue line's length 'dist'
If you compute the arctangent of ratio of the half extents (Y / X), then you can get the angle. From there you can simply linearly add Pi/4 to the angle (45 degrees), then use:
newx = cos(newAngle)*dist;
newy = sin(newAngle)*dist;
Repeat for each of the four points, and then find max/min boundaries of the AABB.
This is super tedious. The better way is to use 3x3 matrix representing rotation and translation. In effect, you take each of the four points, translate them back to origin by subtract the center's coordinates from them so that their new center is on (0,0), then rotate them. Finally, you translate them back to the real center. With the new set of points, once again, simply compute new the AABB for those points. Wikipedia has some great information about matrix math, especially how to do rotations, translations, and more using 3x3 matrices.