Let's clarify- do you want to build a driver, or the whole globe? Do you have a broken driver or no driver at all?
The driver is not so hard. You need a high voltage flyback transformer (that's the tricky part). There are two flavors past that- one's a Hartley oscillator that automatically adjusts to the resonance of the transformer and globe, or a fixed freq 555-type circuit. The 555 doesn't find the resonant point, but actually this isn't a big deal, the globe operates fine without it. In fact, the globe has interesting variations in its streamers as the freq is varies.
I have a PIC microcontroller doing fun things with a plasma globe. It slowly swings from one freq extreme to the other so the globe varies from a faint white glow in the center to a reddish glow on the exterior.