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In general, copters need at least two props, one for lift and the other to cancel out angular momentum, either as a tail rotor or coaxial main rotors. But U Penn have shown that with the right configuration, you can fly a copter with just one motor and prop. From IEEE Spectrum, here's a good roundup on related research. Excerpt:

 Quadcopters don’t have this [angular momentum] problem, because they have an even number of spinning rotors. Conventional helicopters use a tail rotor to counteract it. But if you’re looking to simplify, are there any other options for creating a MAV that’s passively stable, meaning that it can hover in an arbitrary position in space without any control inputs? Why yes, there are:

And thanks to UPenn for being awesome and showing us what happens when their robots don’t work.

By reducing the number of motors and actuators, the price of the robot drops drastically, and its weight goes down and endurance goes up, making it much more versatile. And as with 1STAR, if you want to make a lot of robots, low cost and simplicity become some of the most important design features. 

The obvious question that we have now is, is there any reason why you couldn’t combine these two platforms, yielding a single motor, single rotor flying robot that’s both passively stable and fully controllable in pitch and roll? Maybe there is a reason, but if there isn’t, you can bet we’ll see a prototype from UPenn at a robotics conference sometime soon.