Where To Begin
With the sign unplugged, gently wiggle the neon tubes every few inches to test for breaks. A break means that tube will need to be replaced.
Testing Neon Tubes
If you have a working transformer, you can easily test your neon tube. If it lights when you plug it in, it is working. If it lights, but is weak, the gas is low. Refilling the gas on a neon tube is a complicated procedure and it is probably easier to replace the tube.
Testing the Transformer
If you know or believe the tubes are functioning, test the transformer. The easiest way to do this is to connect it to tubes you know are working and see if it lights. Older style transformers can be tested by disconnecting the leads and arranging them next to each other. When the power is turned on, the electricity should arc between the leads.
Testing Switch and Cord
If you think the switch may be bad, this can be tested by first unplugging the machine and taking apart the switch wiring. The two wires the switch wires connect to should be wired together with a wire nut on the connection. Plug the machine back and retry. If it works, the switch is bad. A test light should be used to check a cord for continuity.