Effectively all you're salvaging is a small fluorescent light bulb. You could try buying an inverter that seems like a reasonable match for the typical bulb size from Digikey and seeing if it will drive each of the tubes, and if so, buy enough for your project of lighting them all up.
Aside from what tube size the inverter will drive, some notable variables will be the DC input voltage range and the connector to the bulb (which is not standardized). You might care whether the brightness control is an analog knob on the board or a digital control signal. There's a lot of variation in board shape and component height (for fitting into different types of products) that may not matter to you.
If the project gets too complicated, you could try salvaging only the diffuser panels and replacing the bulbs with a single bulb part number that you control. That way you can match the tube and the inverter and have control over the whole design. As the inverter cost is vastly more than the bulb cost, you're not saving much money by reusing old bulbs (which are likely already much less bright than when they were new).
For more technical details, you could start with this app note from Linear Tech.