You sure you can't rock the toilet? Maybe only if you lean back or just to one side?
A joint in the drain line can have a small leak if it wasn't primed or glued properly. But, if it is good for a month, it doesn't seem it would be that. Does the floor flex at all when you are walking around the bathroom? Or, does it have a tub? If you use the tub mostly for showers and occasionally for a bath, could it be leaking after the bath? That extra weight if the structure is not great could flex things. You sure it is the toilet? Water can run along the pipes and look like it is coming from places it really isn't. What about the shutoff valve, could it be leaking or the elbow in the wall? Are the pipes copper or galvanized?
These are all WAGs. I'm not a pro, just thinking off the top of my head.