Create a simple flip flop circuit
Here is a schematic for a typical T flip-flop. Every time you press the button, the output switches.



All you need to do is to build the logic gates! You can use diode-transistor logic, resistor-transistor logic, whatever. When you're all done, connect the inverted output (Q#) to the data pin and connect your button to the clock.

You might consider debouncing the switch. This can be achieved using a resistor and a capacitor.