The flash capacitor could be bad, or the batteries could be weak- make sure they're charged, and clean the battery and camera contacts with a qtip and at least 90% isopropyl alcohol.
The flash consists of a small step-up transformer, a capacitor, small ic, and the xenon bulb. Some terminals on some cameras are coated with an insulating glue, but one some you can discharge the capacitor with a screwdriver- be ready for a fairly loud "Pop!". Do it a couple of times, or whatever it takes to fully discharge it. Look for broken, bad, or cold solder joints on the relavent devices. Take the old solder up, and resolder, if required. You can ohm out the transformer, and cursorily check the cap, but caps can break down while trying to charge, so that isn't always a conclusive check. The flash board is usually a separate board, and can be replaced as a unit. You might try to get an "as is" model off Ebay with a good flash circuit, and replace the board, bulb, and cap as a unit, but that's not an easy job. Other alternative is to get a new camera, since if you elect to have it serviced, it will justify buying a new one, in most cases. I found a working replacement on Ebay for $.99, but the shipping is $26.90 (you could try to talk them down). Here's the link:
Good luck, and hope this helps.