I don't know if this problem also exists for HDMI cables, but at least there are DVI cables (which are basically the same), that lack the so-called "DDC"-pins in order to lower production costs, which means that the monitor can't be recognized automatically. Usually that isn't a problem, because you can specify your resolution manually on a normal operating system, but an Xbox might be confused. This is just a wild guess, though.
So, if anyone of your friends has an HDMI cable or a DVI<->HDMI adapter, I'd ask them and just try out other combinations. I have connected my Xbox to a monitor using an HDMI cable and a DVI<->HDMI adapter, and I can select my monitor's native resolution (1280x1024).