Note I'm making some assumptions on the CDI circuit design based on what seems to be common in the industry; there's a thousand ways to skin a cat.
My guess is that the most likely component to fail in a CDI box is the Silicon Controlled Rectifier (SCR). Fortunately this is a device that you can test, and it's pretty easy to test with a multimeter, even with it in the CDI circuit.
1) The SCR anode is connected to the ignition coil and the CDI capacitor
2) The SCR gate is connected to a trigger signal, through a resistor
3) The SCR cathode is connected to the bottom end ("ground") of the primary winding on the coil
Step 1: Put your multimeter in resistance mode (use a high range if using a manual meter)
Step 2: Connect the positive lead to the SCR anode, connect the negative lead to the SCR cathode. The meter should show an open circuit (high resistance)
Step 3: Reverse the leads (positive to cathode, negative to anode). The meter should show an open circuit (high resistance)
Step 4: Connect the positive lead to the SCR anode and the SCR gate, connect the negative lead to the SCR cathode. This should turn on the SCR and show a short circuit or low resistance.
Step 5: While leaving the SCR anode and SCR cathode connected to the meter, remove the connection to the SCR gate. This should continue to show a short circuit or low resistance, showing the latching behavior of the SCR (important).