On my current .blend file, when I edit a particular object's origin point and save it, opening the file again resets the origin back to where it was before. I'm unable to save the object with a new origin location. I tried doing CtrlA to set to apply location, rotation, and scale, and then saving it, but upon reopening the object was scaled strangely and the origin once again not in the right location.
It sounds like you accidentally added a keyframe to your object. So even if you move it, it will snap back to the keyframe position. You can fix this by opening the NLA editor and deleting all animation on your cube, or opening the Dope Sheet and deleting the keyframe you don't want. A common way to add keyframes by accident is by pressing the Auto Keyframe button (at the bottom of the timeline), this causes any rotation, scale, or movement to your object to be keyframed.
You can acsess the Dope Sheet through this menu:
Once in the Dope Sheet, select the keyframes you don't want and press delete.