Here are the simplest 120V LED circuits I know of:
Make sure the capacitor is rated at 200V or more and I think a 1/2 watt resistor should do the trick in this setup.
If you decide to try to get the neon bulb working, here are some thoughts on that:
Measure the voltage between the 2 big wires (around the resister and the bulb) and you should see the total volts in the circuit (assuming the bulb is blown). With the power off you can check if the resister is good and then if the bulb is blown.
First of all, make sure the multi-meter is set to an AC voltage range of at least 120V, not DC. When measuring the voltage between the 2 big wires as suggested by Slowbrew, if you get 120VAC then power is getting to the bulb/resistor combo, otherwise power is not getting that far (i.e. the problem lies somewhere else). If the power is getting to the bulb/resistor combo, carefully measure the voltage across the two leads of the neon bulb. If it reads 120 VAC then the bulb is dead, otherwise I would suspect the resistor (or both).
It looks like Radio Shack has 120V mini-neon lamps - that might be your easiest fix. I can't tell for sure, but it looks like the assembly may include the resistor internally.