The "diode test" is simply a resistor from a voltage source to put current though the diode, with the multimeter reading the voltage across the test terminals.
A battery and resistor are all you need, with 4mm sockets and patch leads if you want to make a really fancy version rather than just hold the meter leads to the same places.
The battery should be something in the 6 - 9V range, so the current does not change too drastically with different LED voltage drops, and there is still sufficient voltage when the batteries are near flat (2/3 of the nominal voltage).
A 1K resistor should be suitable with either voltage, or 470 Ohm for more output; anything less than 20mA should be pretty safe on most things.