You're going to have to at least partially remake the thing. Probably. Unless you managed to avoid all non-storing actions (read - vertex editing), you won't be able to go back and tell the program to add more divisions for faces for the previous results.
This is one area I've run a number of experiments in.

See, the problem you have is that the polygon face that has the subtraction is a contiguous object. If you cut it across the middle of the face, where will the actual geometry that defines the new edge go - will it cover over the hole? If it let you get away with that, I'm pretty sure you'd get non-manifold geometry. So the program protects the user from the problem. In a case like this, unless you can undo the actions to a point befor you booleaned the cubes, you'll need to remake the object from the beginning.

Or wait maybe not. I just thought of this, and I have no idea if it will work. If you still have the object's history, go to the mesh in the outliner that was chronologically there before you used the boolean opertation. click to select it. Now try your cut. I have no idea what will happen, but it's worth a try.

If it doesn't work you'll have to remake it. Yes, bad news. The good news is, as long as you have just one cut somewhere through where the hole will be, you should be fine (If I'm remebering correctly). The program will then allow you to make further cuts. Although personally I'd reccommend starting the cuts about where you were going to put them anyway.