Diodes, resistors, LEDS, globes (it has more things but we are allowed to only use these)
Are you allowed to use a multimeter (or voltmeter/ammeter)??
This would considerably help with designing an experiment because you could measure the voltage and current (and hence calculate power) across components rather than buying some fancy device to measure light intensity of an LED.
If you were allowed these measuring devices, a simple (but hard depending if you're allowed a variable resistor or not) is you could vary the current in the circuit by adding resistors and measure the voltage across the diode (with an LED to dissipate excess power). Then plot voltage vs current and find the switch on voltage
(note that this could be VERY hard if you only have large-sized resistors, so google the diode before hand to see what the Vs is)
Otherwise, as others have suggested, you can create a variety of circuits and analyse. It all depends on what your teacher wants
(Perhaps even find someone who did Physics last year from your school and ask what they did, but don't do the exact some thing
)
GLHF